You are going to do more harm that good with a catty, even with an air rifle, you are going to have to be a top class shot to kill cleanly.
Used one of these, which work, remember I think its still a no no to relocate a live one, so you are still going to have to find a clean way to dispatch them & dont expect them to sit still when you get close 😯
http://www.greenpestco.com/Squirrel-Control/Squirrel-Trap/prod_74.html
It will only take weeks for more to take over the area once cleared. I no longer trap, btw so there might be better methods about now.