[footfootfoot]

And always keeps the tree between himself and the hunter

This is the crucial bit of information.

Consider a bicycle wheel for example. The hole in the rim for the valve stem is opposite the seam of the rim. The wheel revolves around the axle. The valve stem and seam each go around the axle every revolution. They do not go around each other. They do go around the position the other previously occupied however. Compare this to a ratchet wrench. The socket is the axis, the handle goes around the axis, but the handle does not go around itself.
The squirrel always keeps the tree between himself and the hunter. The tree is the axis, the squirrel is the seam on the rim and the hunter is the valve stem.

No, the hunter does not walk around his gun. His gun is not stationary, it moves with him. He may walk around the location his gun used to be, but that is not the same as walking around his gun. Unless he has one of those special mag-lev guns that just hangs in mid air while you circumambulate around it. Otherwise, if it is a normal gun then he will be taking it with him when he walks around the tree.