To weigh in on this rather late in the game - I just killed 4 red ground squirrels at our cottage over the last week. We’re on an island and the squirrels migrate over the spring ice.
We originally were “live and let live” until they got in one winter and did thousand’s of dollars damage. Now we’re “live and let die” (cue trumpets) with a standing order to terminate squirrels with extreme prejudice.
To answer your OP - my son’s Red Rider BB gun from Cabela’s wasn’t capable of killing them from any distance more that about 5 yds. At best it knocks them out of the tree and you have to run over and hope to do a point blank head shot before they scamper away.
A friend loaned me his high powered pellet gun. Sorry, I don’t know the brand. It’s exactly at the legal limit of muzzle velocity before it becomes regulated (in Canada) as a rifle. The pellet gun is pretty much a kill shot anywhere in the top half of the squirrels body up to maybe +30 yds.
Bottom line - definitely get a pellet gun vs a BB gun.
Lastly - to reiterate what others have mentioned, you need to check with your municipality on whether you can fire any gun at all.
Obligatory story: When I was in university my roommate and I were doing target practice at our rental house. We were responsible and didn’t want ricochets to hit anything so we set up cans deep in the garage. We sat in the driveway drinking beers and passing his BB gun back and forth.
After about 15 minutes two local SWAT team vans pulled up to the house and maybe 12 guys jumped out in full body armour carrying M16’s and shotguns. A neighbour called 911 to report “gun shots”. While they jumped out, we shit our pants.
Luckily the cops weren’t too aggressive about things (no screaming at us to “get down”, no guns pointed etc.) After we handed over the gun, by the barrel as instructed, they explained that it was illegal to discharge any firearm in the city limits, even BB guns. They did comment that it was good thinking to set the target up in the garage.They gave us back the BB gun and left us with a stern warning never to do it again.
There were lots of students living in the area and we found out the next day that squad cars had completely sealed off a two block radius around our house while the SWAT team moved in. “Holy shit, did you see all the cops cars blocking the streets yesterday. WTF was that all about?” 