Well, I guess one of those 20ga/22lr combo guns. Use a slug in the 20ga.
I can't imagine there would be anything left of a squirrel after getting shot with a 357.
You would be surprised. I went hunting jack rabbits in New Mexico with my 22 revolver and my two buddies had a 38 spl and a .357. both shooting lead round nose ammo in their guns and it wasn't as bad as you might think. Out of the active duty a couple years later and I'm in the Wisconsin woods with my Ruger Blackhawk .357 loaded with LWC after rabbit and see a squirrel. One shot to the mid section took him out of the tree and all the edible parts were still there. Little tricky to skin but cleaning was easier. It would be easier with a carbine length gun.
Good intentions will always be pleaded for any assumption of power. The Constitution was made to guard the people against the dangers of good intentions. There are men in all ages who mean to govern will, but they mean to govern. They promise to be good masters, but they mean to be masters.