These are not the typical tree squirrels you are thinking about. It’s a different breed of cat, if you will:
And BB guns generally lack the velocity or energy to kill small rodents. I’ve shot tree squirrels with my adult air rifle (Sheridan/Benjamin 5mm (20 cal) pellets) pumped up to about 800 fps and one will go through a tree squirrel but not exit, because the little critters are tough. A 22 cal rifle bullet will work just fine, but that has a danger range of up to 1.5 miles, so you don’t just give them nilly-willy to kids and tell them to blast away.Correct. Ground squirrels everywhere are considered to be destructive pests and may be taken freely.
A better solution is to use the Squirrelinator. It's a no-kill trap that allows you to catch up to 20-25 squirrels at a time. It's a lot like the squirrel motel. They check in, but can't check out.
Once caught they can be dispatched using an optional water basin to drown them or by shooting them through the bars of the trap.
I've used mine a lot over the years and they're very effective.

I invented what I call “The Chipmunk Trap Of Death” in order to save my tomato patch.
I only caught one squirrel, but lots of chipmunks. Squirrels are
generally too smart for this trap. Which is fine because they do not bother
with tomatoes, and I no longer have peach trees or bother with trying
to grow corn, both of which squirrels DO wreak havoc upon. So I have
no beef with the squirrels any longer.
Chipmunks, however, steer clear of my yard if they know what’s good for ‘em!