Here you go. Took me all of 5 minutes to find this, with a google of "alabama fish and game". The department is actually called Alabama Department of conservation and natural resources. (In CA its the California department of fish and game, it's different depending on which state your in).
OTHER GAME BIRDS OR ANIMALS: Rifles using rimfire ammunition or those operated by air. Muzzleloaders and black powder handguns of any caliber. Long bows, compound bows, or crossbows. Shotguns, 10 gauge or smaller, using standard No. 4 shot or smaller. Handguns or pistols. Blowguns using darts propelled by the hunter’s breath only. Sling shots.
Paraphrase(for actual text see
http://www.outdooralabama.com/hunting/season-limits/2007-08 Season & Bag Limits final.pdf ):
Squirrel: 8 a day - 8 in possesion: October 1 - February 29
You can't use anything that casts a beam of light(i.e. Flashlights, lasers) or any type of night vision. You can't use fully automatic or silenced firearms. You must allow your firearms to be inspected, if asked by a conservation officer.
If you are hunting durring Deer Gun Season you must were Hunter Orange(see PDF for specifics)
So it seems that you can hunt with a hangun, just don't carry it concealed unless you have a permit. You might look into what the open carry laws are in Alabama, but I would assume that you are allowed to carry openly while hunting, or else you wouldn't be allowed to hunt with a handgun. Please don't take my word for it though, go do your own research, and if in doubt, contact a conservation officer.