The best tip I can give you is be sure which species your looking at before pulling the trigger. Here's a quick pull from WDFW on the subject.
The Western gray squirrel is classified as a threatened species and cannot be hunted, trapped, or killed (WAC 232-12-007). The red squirrel, Douglas squirrel, and Northern flying squirrel are protected species and can be trapped or killed only in emergency situations when they are damaging crops or domestic animals (RCW 77.36.030). A special permit is required in such situations.
The Eastern gray squirrel and Eastern fox squirrel are unclassified and may be trapped or killed year-round and no permit is necessary. In such cases, no special trapping permit is necessary for the use of live traps. However, a special trapping permit is required for the use of all traps other than live traps (RCW 77.15.192, 77.15.194; WAC 232-12-142).
*********
So basically you can kill the Eastern Grey and Eastern Fox squirrels, and no other breeds. Those are fairly common in cities, so finding them shouldn't be a problem. The thing with squirrels is they are small enough that a pellet rifle will kill them, so you don't have to worry about shooting a .22 in the city. Hollow point pellets are best.
Source(s):
http://wdfw.wa.gov/living/tree_squirrels.html#legalPics & distribution:
http://wdfw.wa.gov/conservation/gray_squirrel/
