I live on 7+ acres; my nearest neighbors live about 10 feet from my property lines. If I fired off anything (even a firecracker), I could probably expect a LEO on my doorstep within about 15 minutes . . . .
You do what you feel you have to do.
If one doesn't already own a gun, getting a permit and the purchase of one; learning how to handle it safely, finding out what the local regulations are regarding the discharge of firearms within x number of feet of an occupied dwelling, etc, can get rather complicated.
This was going to be my suggestion - fill the trash can with water, and drop the whole cage into it. Cervical dislocation may be quicker, but it involves getting your hands on the squirrel; those things are wild animals and they BITE! You want to take him out as quickly and as painlessly as...