Learn English – Where does “at any rate” come from

The following is what I've found on the net about this phrase:

The Random House Historical Dictionary of American Slang lists the original form as "Doodly-squat," dating from 1934. No clue given as to the origin. Doodle means, variously, a fool, a Union soldier, a penis, to cheat, and to copulate. The dic does not list a usage for "doodly-shit" until 1966.

The dic lists "diddly-squat" as a euphemism for "diddly-shit," which does not appear until 1964 (1963 for diddly-squat).

It is difficult to draw a conclusion from all this, except I doubt it has anything to do with squatting in a dwelling.

Dave Wilton, posting in alt.usage.english

I have also found an interesting story about the etymology of the phrase.

(Editorial note: The copied story has been removed because of questionable copyright use. Also, the linked story is a joke and not a true etymology of the phrase.)

Swag as a dictionary entry traditionally has several meanings, including the one you're referencing.

Swag as a slang word has its roots in "swagger".

Swagger: A very confident and typically arrogant or aggressive gait or manner

It refers to a way your present yourself; having style, confidence, etc. Using the term is so popular among young people that it's generated quite a bit of backlash, especially among the internet culture. "Secretly We Are Gay" and the like are simply backronyms that serve to make fun of people.