calhenry:
i queried the ildnr on this type of subject a couple of years ago on behalf of an out of state person coming to the shawnee hunt that we have every year.
the law dept of the ildnr informed me that on private property or federal land (Shawnee National Forest) a hunter may use any cal of pistol or rifle to hunt squirrels. you may not use a shotgun with slugs. you certainly can use a shotgun with birdshot.
on state property, you must go by site regs. most state property is shotgun only. a few are open to .22 cal rifles and i ASSUME your airgun falls into that category as long as it is .22 or smaller.
most of the state property i know of where you may use a rimfire is in the southern part of illinois. chester and jonesboro.
if you go to the il dnr website you may find out site specific information on il. state property. i believe that is also where i found the legal dept.
again, if you hunt private property or federal land you can use any cal. pistol or rifle and any shotgun up to 10ga with birdshot only. this fellow in question wanted to use a reduced load in .22 hornet which is centerfire which is perfectly legal for squirrels as also is a .458 win. magnum.....
i occasionaly use a .32 cal caplock bp rifle to hunt squirrels here in illinois where it is safe to do so. it's a blast!
if you are a squirrel hunter, drop by the small game forum for a visit. i always enjoy other folks views and stories on the lost art of squirrel hunting!
luck!