The only critters I send to a better place are groundhogs.
My home is built on stilts, and those ba%$#^&ds dig holes all around the foundation, and have gotten under the house, chewing wiring, tearing up insulation and such.
I live in the country, so I use a mini 14 to do the deed.
If you live in an area where you're not allowed to discharge a firearm, they do make subsonic .22 rounds that are pretty quiet, and very effective on something as small as a squirrel.
They're a bit more expensive, and not intended for target shooting, but rather for just the sort of problem you're having.
Happy hunting.