I use a .22 lr. I try for headshots, so it really doesn't matter what I use. Even a .22 HP will tear up a squirrel so it's best to shoot them in the head.
After you get good at it, a scope sighted rifle is almost too easy. Try it with iron sights to put a little challenge back into it. I usually hunt them with a .22 pistol, nowdays.
I haven't used a shotgun for small game, other than birds, since I was a kid. I don't like biting down on shot, and a shotgun can really mess a squirrel or a rabbit up at close range.
Squirrel and dumplin's -- YUM