Bait it with peanuts or corn, then drive a few miles away and let them go in the woods ...
Ten days and you will have your problem solved without upsetting your neighbors ... that is until the new squirrels move into the area.
The traps are the best compromise ... if your neighbors dont care ... just get a .22LR rifle.
A cautionary note. You ought to be a good shot with that rifle, otherwise you're sending a .22 bullet out into the air and it will come down, perhaps onto someone's head. Not likely, but possible.