Serving your country while in academia, or before going into academia I am a 3rd year mathematics undergraduate(see my math se account), and I think I am well positioned to go into academia, and I am willing to do anything to make it happen.

I also have a desire to show my love for America, and I can't think of any unequivocal of doing this other than joining the military.

Many of my competitive advantages have come from completing certain milestones before a certain age(i.e. finishing the trigonometry books that Ramanujan used at the age he did them, BC calc in 8th grade, a competitive REU program Freshman year, going straight to the graduate courses in my undergrad).  Thus getting deployed to active duty might break this string.

Has anyone had any experience in this matter?

I am not, of course interested in industry, at all.  I would sooner do math on the streets than go into any other job where I cannot continue what I am doing 24/7.
I think you should really consider what you are willing to do in the military.  Are you going to enlist or apply to OCS?  What is your desired MOS?  If you want to join the military while continuing in academics, this is very field specific.  For example, the Navy/Army/Air Force employs a number of medical physicists and will pay for their masters degrees.  You need to specify exactly what your goal would be while in the military.

"I am not, of course interested in industry, at all. I would sooner do math on the streets than go into any other job where I cannot continue what I am doing 24/7."

If you join the military, you will almost absolutely break your 'string' of math achievements.

"the wars that are fought are antithetical to my conception of American values"

If you are enlisting and you branch infantry, what will you do?  If you become an armor officer, what will you do?

All academics aspire to 'serve' humanity in their own way.  Research pushes mankind forward, not nations.