I've been known to trap squirrels. I used a properly sized live trap and baited it with BOSS (Black Oil Sunflower Seeds). I'd put the trap at the base of a tree (so they are comfortable they have a quick escape route) and scatted a little BOSS around it, then put more BOSS on the trip plate. Using just two traps I once removed about 50 squirrels from my suburban back yard in a month. When I started it wasn't unusual for me to see at least five squirrels back there anytime I looked. As I trapped them new ones would come in from the surrounding back yards.
If I disturbed the soil, either planting something or just pulling a weed they'd dig a hole looking for a nut. And they were causing property damage, mainly chewing up the flashing around my roof vents for the salt in it. There were just too many. I eventually thinned them down to manageable numbers.
It's probably illegal to release them anywhere except your own property and probably illegal to kill them, especially out of season without a license. You might talk to animal control and see what your options are. I took them to an oil refinery out in the country about ten miles away and released them, far enough away that they wouldn't find their way back. Or if I were going fishing in the marsh I'd dump them there. Not legal and not very courteous to people that lived around there but hopefully the gators were well fed. I wasn't ready to handle that many squirrel bodies.
It sounds like you are targeting a certain squirrel so I'll tell these stories. In my 50' x 75' garden in Arkansas one spring, a rabbit was eating my bean sprouts as fast as they were coming up. I killed 16 rabbits out of that garden before I got the last one doing that. That was rabbits I saw in the garden, not just ones in the area. I don't know how many of those rabbits were actually eating the sprouts, probably not all of them.
Also in Arkansas a skunk came through a pet door to get in the garage and sprayed when the dogs noticed it. I was not pleased. It was breeding season so the males were on the move. I killed seven skunks in the next couple of weeks. I don't know if I got the guilty one or not. If you have one squirrel, skunk, or anything else you have more.