I carry a "possibles bag" slung over my shoulder with:
A pair of needle nose pliers
A pack of Kraft paper sandwich bags
A pack of Handi-whipes
A pack of surgical gloves
A goatskin tobacco pouch full of .22 shells
Hanging from the shoulder strap loop of the bag is a 2 foot length of cord, and I have a VERY sharp pocket knife.
I carry squirrels by looping the cord around the neck and letting them hang down like a stringer of fish.
When ready to skin them, I tie the cord to a limb or a stout nail in the barn. I cut through the squirrel’s tail at the root, leaving it attached to the body only by a strip of skin, and use a slip knot to hold the squirrel hanging.
Then I continue the cut made in the tail around the body, leaving a V-shaped cut on the abdomen, pointed toward the head. I grab the hind legs and pull down until only the head and front feet are still encased in skin.
With the needle nose pliers, I grab the V of skin and pull down, and continue pulling until only the hind feet are encased in skin. With my knife, I sever the head, then clip the front ankles with the wire-cutting section of the needle nose pliers.
Holding the squirrel by the skin from the back end, I slit the belly and dump out the organs. I hold the gutted squirrel in a standing Kraft sandwich bag and cut the rear ankles with the needle nose pliers.
Of course I wear surgical gloves throughout the process, and frequently wipe hands, knife, pliers and so on with the Handi-wipes. I never touch meat with anything that has even a single hair on it.