I have killed a truck load of squirrels with 177 pellet rifles. Most fell to a 2100 Classic at ten pumps shooting a Crosman wadcutter. Catch one sitting up and hit him square in the chest with a wadcutter at 550 fps, and you get to decide how to cook him for dinner.
The advice that where you hit is more important than size is true. It doesn't matter at all if you hit just behind the eye with a fast bb or a 22 HP. If you wait till he sits up and angle a pellet thru the vitals, a 177 works as well as anything.
In the smaller calibers your margin of error is less than with the bigger bullets. I have a Techstar in 22 sitting and I am tuning on a Cannon Pumper rifle in 177 for squirrel hunting this year. I am setting up a self contained squirrel sniper rig. Once you get the 177 stepping out there over 800 fps, the only question is bullet placement. At 900 fps, solid hits on squirrel heads produce clouds of red spray out the other side. Dead is dead.
Rabbits are easy to kill. Squirrels much tougher. Possums and coons are about the same. Ground hogs are harder to anchor in place than any of the others discussed.
Reasonable range and accurate placement trumps caliber and FPS every time.