I've been successfully killing squirrels cleanly with 6-12fpe guns for going on 30 years. Almost all of those were .177s.
The argument of "I've had them run off" can also be said for a round nose .22LR at 98fpe. Been there and done that too.
A small caliber air rifle does not kill by shock or whatever other word you want to kill it. It's penetration, plain and simple. The fact that they can also run off due to a poor shot with a .22LR proves this to me.
At 35 yards, a .177 leaving the muzzle at 12fpe will pass clean through the vitals of a squirrel and be through the far side, or just under the skin.
I don't know when squirrels got armor plated and unkillable with anything less than 32fpe, but it sure hasn't happened in my lifetime.