I hunted squirrel and rabbit a LOT when I was younger, as did most of the men I knew when I was a boy. We ALWAYS gutted them as soon as they were killed ... just slit the belly open with a pocket knife from the crotch to the breast bone. Then we just reached inside, grabbed the stuff and pulled it out. If we were hunting with dogs, the dogs just gobbled up everything, otherwise we left the innards on the forest floor for other beasts to eat. Everyone I ever hunted with did the same. We waited til we got home to skin the animals out, but we always gutted right on the spot ... same with deer, bear, moose, etc. With the larger animals we were careful to cut around the anus so the entire intestinal tract could be removed without contaminating the meat. We believed that leaving the guts in place made the meat taste strong. The hide was left on to keep the carcass clean until we could deal with it.
I had a commercial rabbitry for about 5 years and slaughtered and dressed rabbits for sale in supermarkets. The routine there was to first break the neck quickly with the "poacher's stretch". Then hang the animal by a hind leg and cut off the head with pruning shears, letting it bleed freely. I would hang about half a dozen at a time. Next I grabbed the skin in the middle of the back with both hands and pulled in opposite directions. Rabbit skin is soft and tears easily, and can be pulled off toward the head and tail like a glove until you get to the feet. The feet were then cut off with the pruning shears, taking the hide with it. Next the belly was slit and the intestines and lungs removed. I saved the heart, kidneys, and liver. After the carcass was washed, the organs were placed back in the cavity and the whole put in a plastic bag.
Squirrel hide is a bit tougher. We generally hung the gutted carcass by a hind leg, then slit the skin along the inside of each hind leg down to the crotch, and slit around each rear ankle. Then the skin could be peeled off just by pulling toward the head. We cut off the feed and head, taking the hide with it. Although I hear of folks eating squirrel brains, we did not eat any part of the head.
Cheers ...... Charles