I've seen former medics/PJs/SARCs become nurses, but haven't seen too many new grad BSNs become medics. While a BSN would give one a solid academic prep it's not a "natural" transition given that > 90% of nurses are women. Not unheard of, just unusual.
The foundation of all of those programs is rooted in emergency medical care, and you would be wise to take an EMT basic course, if not a paramedic course. There is a philosophical difference between pre-hospital care and nursing.
I was a corpsman for a bit after I got my BSN, which did give me a leg up at promotion time and for some schools and billets, but I didn't use 90% of my nursing skills, whereas I did use 90% of my skills that I learned as a paramedic.
There are some very cool operational billets for nurses in the military, it's not all in a hospital. There is enroute care (i.e., flight), SOST/Special operations medical support, all sorts of neat, niche units. When I got my commission in the Nurse Corps I navigated towards enroute care which was a natural choice for me, but in hindsight wish I'd spend some time with FRSS/STP.