Go to the website of the CA fish and game commission LINK
to learn the legal requirements and season for taking squirrel. Apparently only tree squirrel species are legal but not ground squirrel species. You can't hunt in all counties or most municipalities. You must have a small game hunting license and cannot use a pellet pistol. I don't think a squirrel can be consistently and humanely killed with a 425 fps, .177 cal pellet pistol which is probably why the regs exclude them.
Beyond the legal aspects, there's no reason not to eat squirrel except that they offer very little meat and are pretty stinky little critters to clean.