Is it legal to go squirrel and morning dove hunting with a bb gun?
Even if it isn't actually illegal (which it probably is), you should not do it. A BB gun is not powerful enough to kill squirrels and Mourning doves outright, so a lot of them will go off wounded, which is very cruel, and may make you liable to prosecution by your local ASPCA.
EDIT(JACN85): This is only true however for low powered guns or an inexperienced hunter.
I suggest if you do decide to hunt doves or squirrels, to please practice before you do attempt to hunt. Most guns that have a pump action need to be pumped at least five times to effectively kill a dove and even higher for rodents. Also, learn the different types of ammunition used. I would suggest just using Destroyer Pellets as they can handle both prey easily. Never use regular BB's on rodents, the chances of them dieing is very small, please use hunting pellets or destroyer pellets.
As with any sport, practice it safely. Always have your firearm on 'Safe' until right before you fire, barrel pointed at the ground when walking.