.30 caliber for squirrels? Why and why not?

My ma has been liking to come over with nicely filled bird feeders and see what all new wild birds she can attract and view. Squirrels are wearing it out as the feeders aren't squirrel proof. I see it as much better squirrel dumplings. Every squirrel for a few square miles has feasted. My pitbull has snagged two on their way up a tree trunk. He's become quite stealthy.
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Have a couple troublesome coons. I hear they can rip a dog up pretty good.

Coons pretty sure they are the best qualities of cats dogs and squirrels