How long before you early season squirrel hunters skin them out and get them on ice? I imagine a 4 or 5 hour hunt might leave you with some bacterial build up.
I skin them as I kill em and wait for another to show.
Put em in a Ziploc bag in my vest. Usually only hunt a couple hours when it is warm.
I have at times taken a tray of ice and dumped it in the Ziploc that I carry. As it melts I have a little water to wipe my mitts that may be grubby from skinning and gutting.