I think most people have it backwards. Being a hunter makes you a better soldier, or at least an infantryman. I was an infantry EM and officer and a lot of what I learned in the woods hunting helped me, especially in the area of patrolling.
As far as shooting, I think the fact that I was a good shot before I went in helped me. I qualified Expert with mypersonal weaponall but one of my fourteen years in uniform. But I have taught people in the militarywith little shooting experience to shoot as well as I did.
Moving through the woods without a lot of noise is something that most people without a hunting background can't do. Most of it can be taught to a soldier, but there is a reason why the military spends a lot on commercials that appear on outdoor programs.
As far as being a soldier making you a better hunter, maybe you estimate range better, move quieter, camoflage better, etc.... I won't say that it is a given.
But, I would say you have a better chance if you were infantry and not infantry support (Navy, Air Farce, REMFs).