Learn English – Origin of the word “jack” to mean theft or to steal

I guess your question is more about the jack part.

In English a jack is by-name for a common person.

In British English, jack is a very old (13th century) term to designate the average peasant - the man at the bottom of the social pyramid. See for instance colloquial expressions such as "every man jack".

In that sense it comes from Old French "jacques" which has the same meaning - "Jacques" being a very common first name in medieval France at the time. The revolt of the French peasantry during the Hundred Years' War was famously called the "Jacquerie" because the jacques were all in arms and busy burning castles.

It also gave English the word "jacket" that was then adopted back in French as "jaquette" (the lost "c" and the meaning of a typically classy suit is a tell-tale sign it does not come directly from Old French).

During the British naval supremacy period, jack was also used to designate the average seaman.

The word must have somehow passed into American English. See for instance "lumberjack" for "lumberman".

So in addition to the word "shit" symbol of something of little value, the use of jack here reinforces that meaning by referring to an average fellow of supposedly low level of sophistication or knowledge.

Another very close way to see things is that "jack-sth" is used to denote a smaller version of this something. The OED says:

applied to things of smaller than the normal size; [...] jack-bowl, jack-brick, jack-fish"

So you get the idea: a jack shit is of even less value than a regular-size one (who can claim now that EL&U is not an instructive forum ?).

To redeem myself, let me just mention one of the proposed etymologies of the name "Union Jack" (quoted from the OED) as this is a related matter:

"A ship's flag of smaller size than the ensign, used at sea as a signal, or as a mark of distinction; spec. the small flag which is flown from the jack-staff at the bow of a vessel (formerly at the sprit-sail topmast head), and by which the nationality of a ship is indicated, as in British jack, Dutch jack, French jack.    In British use the jack has been since the 17th c. (except under the Commonwealth) a small sized ‘Union Flag’ of the period (Union Jack), which has also been, since 1707, inserted in the upper canton of the ensign; hence, the name ‘union jack’ is often improperly applied to the union flag itself, when this is not carried or used as a jack. Every maritime nation has a jack of its own; this is usually, either as in Great Britain, †the German Empire, Sweden, and the United States, the same as the canton of the ensign, or, as in France and the Netherlands, identical with the ensign, only smaller". (Prof. J. K. Laughton.)

A Google Books search finds multiple instances of shill in the sense of "accomplice" from Robert Brown, "The Watch," a short story set in an auction in New York City, in The Metropolitan Magazine (April 1911):

The auctioneer passed the cheap glasses into the audience, looking after them fondly and retaining the leatherette case in his hand. "You are now looking at what is without doubt the finest, most beautiful piece of workmanship you have ever held in your hand. Genuine Lemier! You all know what that means! Why, gentlemen and ladies, it's the same as the word 'sterling' stamped on silver, or '18 caret' on gold. It's money in your pocket if you buy this. It's an investment. Every gentleman must have have one. I'm bid dollar only"—he took a bid from one of his "shills" (another name for bid-boosters) in the audience.

...

A derisive laugh went up from the well-trained shills [at the auction] and Dougherty condescended to smile, for deep down in every man is a fondness for the stable article, for the fine thing that will last a lifetime, not for the showy fashion to be discarded to-morrow.

The shill beside Dougherty handed the watch back to to him at that psychological moment. ...

"You'd start it [the bidding] for a trifle—five dollars, surely?" prodded the auctioneer in a manner so evident that the shills themselves would have blushed at the imposition, were they not as case-hardened to that kind of crime as the auctioneer himself.

...

Dougherty glanced at the shill opposing his real money with mind money. Let that tall, pasty-faced gink beat him, get that little gem of a jewel of a watch that belonged to Mame—never!

The earliest Google Books match for shillaber, meanwhile is from Word-Lore: The 'Folk' Magazine, volume 1 (1926) [combined snippets]:

In this case an old circus trouper is the speaker, and his remarks are taken seriously, and a circus "glossary" is provided. (This glossary, by the way, includes the following curious item:—

Shillaber—An employee of the circus who rushes up to the kid show ticket box at the psychological moment when the barker concludes his spiel. He and his fellow shillabers purchase tickets and pass inside and the crowd of towners in front of the bally stand are not slow in doing likewise.

With which lucid explanation of course we know all about the shillaber. But the glossary is silent as to barker and spiel).

Earlier than that instance is one from "Grafters Go Glimmering," in the Tonopah (Nevada) Daily Bonanza (May 29, 1920):

Tonopah will have a sane and sensible Fourth of July without the raucous barking of the shillaber of the street carnival. The people of Tonopah expressed themselves in no unqualified tone when they learned what was contemplated for the national holiday week and forthwith sent up such a protest that the American Legion refused to be a party to shielding the fakers and dropped all negotiations. Merchants on Main street refused to sign a petition for use of that thoroughfare and the citizens of Brougher avenue turned a deaf ear to all appeals from the circus crew for consideration.

And from a full-page ad for "Big Auction Lot Sale," in the [Ardmore, Oklahoma] Daily Ardmoreite (May 16, 1920):

GUARANTEE[:] A certified check for $5,000 has been put in the hands of the Mayor of Grandfield as a forfeit to be paid to anyone who will prove that that there are any by-bidders, cappers or shillabers on the grounds. Absolute aution sale and everything above-board.

And earlier still, from W.W. Chapin, "Success: San Francisco's Symbol," in the San Francisco [California] Call (January 1, 1913):

It is the popular thing these days for the smaller cities to adopt what they call a "slogan," slogan being the battle cry of a barbarian, the shibboleth, as it were, of a shillaber. "Watch Bugleville Grow" they cry. "See Squashville Swell," or "Bungtown Booms." Amusing as such war cries are still their uses are like those of the bass drum of the Salvation Army.

Anatoly Libermn, "Extended Forms (Streckformen) in English," in Studies in the History of the English Language II: Unfolding Conversations, cites The Oxford English Dictionary as having an example of shillaber from 1913:

Another word with stress allegedly falling on the first syllable is shillaber (American slang), of which the OED, The Second Supplement, has only one 1913 example. The lexicographers on the present staff of the dictionary could not have heard it, for it has been dead for almost a century, and only its doublet shill 'decoy or accomplice, especially one posing as an enthusiastic or successful customer to encourage other buyers, gamblers, etc.' is in use. Shill is probably an abbreviation of shillaber (so in the OED), whose origin is said to be unknown. The development from shillaber to shill does not prove that the longer word had initial stress: cf. prof from professor and, conversely, 'burbs from suburbs. Could shillaber be an extended form of Germ. Schieber 'black marketeeer' (*shi-la-ber)?

But unless the OED's instance from 1913 is the item from the January 1, 1913, San Francisco Call, the latter probably antedates the OED's example. In addition to this item, the Chronicling America newspaper database finds two matches from before 1913. From "Blacks May Get Together Next," in the [Grand Forks, North Dakota] Evening Times (September 12, 1910):

"The next big fight for the benefit of the Shillabers will be between Jack Johnson and Sam Langford. Don't overlook this. This match isn't so far away as you may think. I have it straight that Mistah Johnsing isn't doing so well in the theatrical line and I also know hat his financial state isn't any too healthy. ...

And from "Claims Men Are Fakers," in the Los Angeles [California] Herald (February 15, 1908):

Despite the warnings and repeated arrests by the prosecuting attorney, Edward J. Fleming, the barking peddlers of phony jewelry and jejeune junk, with other alleged peace disturbers, including cane-and-ring and similar rack gamblers, continue to ply their graft at Fifth and Main streets, in front of the Beaumont cafe, and also at Fifth and Los Angeles streets.

...

Frank Perkins, it is alleged, conducted a game in which the players tossed rings at silver money. Simpson and Dwyer are alleged to have been his assistants, known as "shillabers."

Also of interest is an item from 1915 that explicitly connects shillabers and shills. From "Says Summer Park Games Couldn't Be Beaten," in the Chicago [Illinois] Day Book (October 29, 1915):

The noble art of bunco—like tickling a man and picking his pocket—as practiced at Riverview is told by Jacob Le Bosky, attorney for some concessionaries of the park, in an action to investigate a receiver's report.

...

"The paddle wheels where candy and various other prizes were distributed with ever-r-y turn of the wheel were money makers. Some of them had 'shillabers' or 'shills,' employed by the owner of the concession.

"The 'shill' would buy a paddle and stand in the crowd with the rest. The man operating the wheel knew what numbers the 'shill' had on his paddle, so that when the wheel was turning, by means of a squeeze attached to the machinery, could stop it on a number which gave the prize to the 'shill.'

Conclusions

The term shill in the sense of "accomplice" goes back at least to 1911; and the term shillaber in the same sense goes back at least to 1908. An article published in 1915, when both terms were in active use, explicitly describes them as alternative words used to describe the same thing.

Usage seems to have spread fairly rapidly across the United States. By 1915, instances of shillaber and/or shill had appeared in periodicals in Los Angeles (1908), New York (1911), San Francisco (1913) and Chicago (1915)—but also in Grand Forks, North Dakota (1910).

Unfortunately, none of the early matches for shillaber gives any hint of where the term came from and what earlier word or person (if any) the term is based on. A New England humorist named Benjamin Shillaber had invented a character named Mrs. Partington, who became famous for such remarks as that she never worried about the price of flour because whenever she bought 50 cents worth, it always cost the same—but Shillaber died in 1890, and he seems to have contributed nothing toward the bunco arts.

Leo Rosten, who isn't shy about asserting Yiddish influences on U.S. English speech patterns and vocabulary, makes no claim for an association between shill or shillaber and any Yiddish word. I think this is another case where "origin unknown" continues to be the safest call.