In spring and early fall, a scatter gun is your best bet. With all the leaves still on the trees, it's tough to track them running from limb to limb through a scope, or even with open sights. Once the leaves fall off, hunting them with a .22 will make you a better marksman overall. Personally, I don't care to hunt anything with a shotgun that I plan to eat, because I don't like biting down on steel shot

However, I'll use one when my .22 isn't the best option.
A smaller-bore scatter gun will usually server you better (20ga is about perfect), but if a 12 is what you have, use it! #7 shot will do the trick, but I suggest a modified choke. If you're a good shot with a full choke you may actually get too many pellets in them. That's just my personal preference, and a full choke will work fine.
I don't know how the laws are where you are, but here in AR we aren't allowed to hunt any small game with a centerfire rifle - only rimfire. I'd check your regs before using anything else. The .17s are too hot for squirrels IMHO if you plan to eat them, however a head shot doesn't mess up any meat

Unless you're one of those weird folks that eat the brains (yes, some people do!)