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This use of book is definitely earlier than the 1990s. Two posters on Ask MetaFilter date it to the 1960s:

• “Booking it” has been slang for running away (from trouble or a tight situation) for as long as I remember. I am talking late ’60s Boston …
• Thirding (or something) the not-a-Mass thing, it was in wide use in California in the 60s.

Some 10 remember it from the 1970s, and around 15 from 1980s.

From The Phrase Finder:

A couple of references associate “book it” meaning to move fast with “book it” meaning to study or “hit the books”. (Black Talk: Words and Phrases from the Hood to the Amen Corner by Geneva Smitherman, Houghton Mifflin Co., Boston, 1994; and Random House Historical Dictionary of American Slang, Volume 1, A–G by J.E. Lighter, Random House, New York, 1994.) Random House says “book it” is influenced by “boogie”. Both sources say the phrase dates back to the 1970s. That’s when I first heard it.

Dictionaries

The first citation in the Historical Dictionary of American Slang is from 1974:

Time to book this joint.

Another edition of the same book from 2002 or earlier links it to boogie:

book …
3. [infl. by BOOG, BOOGIE, v.] to leave; to go fast; move along. – also constr. with “it”, “up”.

The Routledge Dictionary of Modern American Slang and Unconventional Langauge (2009) by Tom Dalzell and Eric Partridge agrees on US 1974:

2 to depart, usually hurriedly US, 1974

• Belly sprang to her feet. “We gotta book – fast.” – Seth Morgan, Homeboy, p. 66, 1990
• MARSELLUS: Whatch got? ENGLISH DAVE: He booked. — Pulp Fiction, 1994
• We gotta book it if we’re going to make it to P.E. — Clueless, 1995
• We gotta book. We’re catching a bus to Chi-town. — Chasing Amy, 1997

All the citations and reminiscences are from the US and mostly amongst schoolchildren. I’ve never heard it in the UK, where scarper might be used instead.

alt.usage.english anecdotes

Over in Usenet’s alt.usage.english, there’s a 205 message discussion from 2002 on this (and in 1999). Highlights include a poster (rzed) hearing it in 1968:

I heard this usage no later than 1968 (from a co-worker in a job in left [sic] in that year), in the US Midwest. I don’t know the origin, although I’d think it is at least related to the use of boogie to mean “go”. It may be more common to say something like “let’s book it” than “let’s book”.

Another poster from the mid-west also remembers it from the late 1960s:

I remember hearing the term in the late 1960s. I recall it as a slang term for “to leave”, as in: “Let’s book”. I don’t recall any connotation of running away, but there was a certain amount of alacrity associated with it.

And:

We talked about this before (I looked it up – it was in 1999), and most of the people who knew the usage came from the Midwest. I know someone from Binghamton who says book meaning “move fast”, and when we discussed it in 1999, someone from Buffalo or Rochester also said they used it.

Another:

In my experience, it was universal among U.S. servicemen in Europe in the late 1960s. It meant “to leave”. Period. After about 20 minutes in a bar, someone would yell “Book!” and all the cool people would proceed to the next bar.

And from the same poster in 1998:

In the members of the US military of the 1960s, and its surviving human tatters today, “to book” does not mean “to hurry”, but rather “to leave the premises abruptly”. “Let’s book” means “Let’s get out of here”. At the end of an all-night party, some revelers had crashed (i.e. fallen asleep here and there on the premises), and others had booked (i.e., disappeared).

More from 1960s:

This is exactly the usage I remember from the Calif. Bay Area in the 1960s – “he was bookin’ ” or “they were bookin’ right along”. No other forms of the verb. Now I’m wondering if “Let’s boogie” has any connection, because it meant sort of the same – Let’s get going. But I thought the sense there was “Let’s dance”. Was “bookin’ ” ever used to mean dancing?

Some suggested origins are:

• booking out of a hotel is to leave;
• from boogie, that also meant to move quickly, to get going;
• you could say of a fast moving car “it booked”, perhaps tied into breaking a speed record, which may be entered in a book;
• booked, as in “departed”, comes from the days of ocean liners – “booked passage”.

Since it's slang, it's going to be hard to pin down the exact origin of the phrase. I can think of two plausible derivations:

1. The skin of a bruised eye is going to be swollen and pulled taut, and thus more reflective and "shiny" than the surrounding skin. I think this is secondary.

2. The most plausible metaphor: Shoe shiners are not as common now as they were in the early 20th century. "Shiner" could refer to how the unfortunate person hasn't gotten their shoe shined, (usually with black shoe polish,) but rather the eye.