Well my first recommendation is to read Washington State game laws. Washington state has lots of protected species. And if they are not that. Then they have a specific hunting season (and even hunting hours).
As far as squirrel hunting. The only two you can hunt are Eastern Gray squirrels and the California ground squirrel.
Eastern Grays are most commonly found here in the city. So you most likely won’t be hunting them with any powder burner firearm. Also we do have Western Gray squirrels here but they are protected. So you will need to be able to identify the two. And know your quarry.
As for the California ground squirrel. They live in the eastern side of the state. And mostly on private land. I have hunted them in an apple orchard. But with an air rifle. Using a powder burner firearm would not have been safe. Also as for California Ground squirrels. There is some concern of them carrying the bubonic plague. So eating or even handling should be done with caution.
California Ground squirrel taking with .22 cal high power air rifle in Yakama. On private property.
Eastern Gray taken at undisclosed location, Puget Sound area. Removed by landowners request. Gun is a low power .177 air rifle.