A Beeman pellet rifle called the R-7 is bad medicine for backyard squirrels. I have cleaned out one hundred pest and vermin squirrels that were not afriad of mankind. Now I have a balanced population of wild squirrels that live in the trees, not my house.
There is also a handy dandy little item that looks like a granola bar. They come in a six pack from the hardware store. Squirrels cannot resist chewing on them and then wander off to die. Less sporting but just as effective as the other method of control.