We also "had" a squirrel problem...
So far this year, I've permanently removed 28 (as of yesterday) of the furred rats out of our pear and pecan trees.
I started out this spring with a cheap-ish Daisy .177 springer (with scope) from Wally World. The rifle has high velocity and is accurate, but it also has a horrible 9 lb. trigger...takes a lot of concentration to get good shots. I got 19 kills with it before a friend loaned me his Beeman R9 (.22; also a springer). With hollow point pellets, it's a FAR superior piece o' work for killing varmits. For the 30 - 50 ft. shots I'm making, the .22 has much better knock down power. The Beemans (especially the R7 and R9 models) are mid-priced, super accurate, have a VERY nice trigger, etc., etc...just a pleasure to shoot.