I'm thinking of hanging a winch from a tree limb. Will the typical Harbor Freight winch hold up outside through the winter?

My first thought is no- but 12 volt ATV winches handle the elements really well so worth a try. There's a contingent of people that would never use a winch as a hoist- I do it all the time without issue.

I bet it would do a lot better if you had a box around it that shed the rain , ice and snow

I have been thinking of putting a pole just outside the garage to hang ene from , using a big rubber mud flap and some wood ends to make it enclosed over the top and on the ends probably wouldn't even need the ends.

just as you go to hang it drill some holes in the mud flap silicone and washers on both sides put it between the mounting pipe and the unit so that it sheds the water a foot in every direction from the unit then put a bucket upside down on pole to hang the remote in when not in use

This would just be for skinning. I was thinking similar.... Build a box around but then I am concerned about red squirrel moving in and tearing it up.

the big mud flap might be enough protection but not so much as to make a red squirrel house the unit is only about 8 inches wide and 16 long a big mud flap or cow mat or truck bed rubber mat , old conveyor belt might work very nicely.

if you have access to sheet metal and a good way to bend it that might work well also with a box I would be thinking completely open bottom no good place to get in and make a nest