When using shotguns, for squirrel and rabbit, I prefer either my Winchester Model 12 or my grandfather's Ithaca Flues model double. #7 1/2 or 8s for squirrels and #6 for rabbit. Lead shot only.
The trick for squirrels is to "bark" them, in other words, aim just a bit off, so they are only hit by the edge of the pattern. Otherwise, you get a lot more lead than meat!
The shotgun is not the best thing for squirrels, a .22LR is much more efficient. For rabbits, the shotgun does a bit better efficiency-wise, but a .22 is still a good choice, unless you are shooting running game, where the pattern of the shotgun gives it an edge.