I have heard (read? where?) that Jimmy crack corn, and I don't care! refers to making corn alcohol! and that crackers also refers to small farmers who would crack or ferment their own corn.

Helen, With a bit of googling I came up with a number of different interpretations.

Jimmy crack corn is said to come from:
--the practice of "corncracking" or grinding dried corn for use as grits and meal;
--the old English term "crack," meaning gossip, and that "cracking corn" was a traditional Shenandoah expression for "sitting around chitchatting."
--cracking open (not "brewing") a jug of corn liquor

My own guess is the first. Rationale: even today, "cracked corn is the prime ration for commercial hogs in the United States", so I suspect that "cracking corn" was a routine and familiar job in earlier farm-life.

BTW, it's thought that the author of the song (published in 1846) was Daniel Emmett, the fellow who later wrote Dixie.

cracker as a derogatory US term for rural white southerner ("used for whites the way the N-word is used for African Americans") is suggested to have come from:

--a Celtic word meaning braggart or loudmouth (Shakespeare; King John: What cracker is this same that deafe our eares with this abundance of superfluous breath?)
--the practice of "corncracking"; thus, a cracker is somebody who can’t afford any other food.
--the sound of whips used to drive cattle and oxen (as in crack the whip)

http://homepages.rootsweb.com/~fcc/main/what's_a_cracker.htm
http://www.straightdope.com/classics/a981030.html
http://www.motherearthnews.com/askmother/jh.shtml