I've had problems with squirrels tearing up my bird feeders for years. I bought a cheap Daisy 880 Powerline at WalMart for ~$30, and it kills the hell out of squirrels. It's a little noisy, but I found that if I shot from inside the house through an open door or window, that deadens the noise quite well.
BUT, after I'd offed a substantial number of them, I found out that not only is it illegal as hell to fire a pellet rifle in town, but here in CO squirrels are small game animals, requiring a small game hunting license, there's an established hunting season for them, and a daily bag limit as well. So I was essentially poaching the little rats out of season and doing it inside the city limits, not to mention inside a residential neighborhood, and the stress of hoping to God some citizen walking their dog didn't see one flopping around in my front yard. Made me nervous.
So, you may want to consider any possible legal ramifications to killing squirrels in your yard, it may not be cool to do so. Once the light came on it my head, I haven't done it since, just ain't worth the risk and stress to me. I started harassing them with a Super Soaker, which got their attention, and finally just moved the feeders where they can't get to them from the tree, and that seems to have solved the problem.
I've also considered checking out a high-quality airsoft rifle, something that would sting the living snot out of them and leave a nasty bruise or welt, but not actually kill them. They're not stupid, and are highly danger-sensitive. Make their visits painful, dangerous and unpleasant, and they'll go elsewhere. I've wondered too about finding or making something non-lethal in .177 caliber to put in my Daisy, like rubber pellets. Haven't found anything.
But if you're in an area where you can pop them without fear, a cheap Daisy 880 Powerline works fine! Don't bother with the crappy scope though, either get a decent one or just use the irons.