In TX, squirrels are game animals and may not be shot with an airgun--like any other game animal. Some counties have seasons, some have bag limits, others allow shooting year round and have no bag limits.
My county has no closed season but has a daily bag limit of 10. In the county where I work, there is the same bag limit, but open season is Oct. 1-Feb 6. and the month of May.
All that to say--you're going to have to check your state and local laws. It varies...
Try here--it seems to be a hefty download
http://georgiawildlife.dnr.state.ga.us/assets/documents/Popular Guide.pdf
GEORGIA
2005-2006 HUNTING
SEASONS & REGULATIONS
www.gohuntgeorgia.com
From what I can see it appears that there is a daily bag limit of 12 and the season is from August 15 to February 28. Page 14
Squirrel Statewide August 15 - February 28 12 per day
And, wonder of wonders, airguns appear to be legal for squirrels! Page 7
SMALL GAME & FURBEARER FIREARMS:
Rifles and Handguns: Any .22-cal. or
smaller rimfire, air rifle, or any
muzzleloading firearm (scopes legal).
Shotguns: Any having shot shell size 3.5
inches or smaller in length with No. 2 or
smaller shot.
You should probably do the download or get a hard copy. There are some special regs for different areas.
It's also worth noting that there is usually some kind of allowance for land-owners to elimate pests on their own property. I didn't find anything along those lines, but a call to the game commission might be productive.