If you get young squirrel they can be fried like chicken and taste great. One of the best meals I can remember was at my Grandma's when I was a pup. Fried squirrel, cathead biscuits with squirrel gravy. Homemade ice cream for dessert. I kept asking which one was the one I shot. Of course you need a "mess" of them to make a decent meal. The older ones are better suited for stew, dumplings, etc. Anything that tenderizes them a bit.
I actually knew an old fella that trapped possum and fed them out on corn for a while. He claimed it reduced the greasy problem. He said all he needed to do was place a 55 gal drum out with a board leaned up to it. They'd climb up and couldn't get out when they dropped in. I've tasted it before, kinda like pork. I'll probably pass on it again.
My favorite recipe for possum is to place it on an oak board, cover it with potatoes celery, peppers and carrots. Bake for 3 hours at 350deg. Then throw everything away and eat the board.