Yes, all kinds of critters eat at sheds. I had a nice little 8 point rack mounted on the front of my hunting cabin. I have seen squirrels and mice gnaw on it and even birds peck at it. It took about 2 years to eat it down to the skull. Probably would have gone faster if it was on the ground.
I had an old-timer tell me " they eat-em for them minerals in em."