Legally there are not many opportunties to hunt squirrel. Here is a little definition I pulled up off the web.
Because legal status, trapping restrictions, and other information about squirrels change, contact your WDFW Regional Office for updates.
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).
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So basically you can kill the Eastern Grey and Eastern Fox squirrels, and no other breeds.