You can snare them easily if you find where they are entering the house (usually in the soffit under the gutter). High # test flourocarbon fishing leader or even a light wire works well. If you have a fence, you can take a 2x4 and lean it up againest the fence like a ramp. Squirrels love running stuff like this. Just make about 20 of those snare loops.. he'll lasso himself and hang. If you make one on the house though.. just check it often, espeically if the neighbors have a full view of your house....

Your .22 pellet gun is a good choice. I use a regular .22 with .22 CB longs or shorts. Out of a rifle they are in my opinion even quieter than a pellet gun, and they hit very hard. Head shots are always best, but you can get away with body shots using a .22 cal rifle or pellet gun.