WA state web site:
Shooting tree squirrels may be helpful if a small, localized population of introduced species is problematic. For safety considerations, shooting is generally limited to rural situations and is considered too hazardous in more populated areas, even if legal.
The Eastern Gray Squirrel has been invading WA since 1925
http://en.wikipedia.org/wiki/Eastern_gray_squirrelMy experience is that even if it is legal to shoot, the neighbors don't want to hear it. And at night, the wife does not want to hear it.