Which Engineering degree field should I be applying for to help my chances of fulfilling my military desire and also help me with employment with NASA after my military career?

My background is astrophysics and software engineering, so I'm not necessarily the most qualified to answer this. But I did spend several years doing graduate work (in astrophysics) at NASA Ames, and I've hung around with a lot of engineers both there and in college.

Aeronautical engineering would certainly be one obvious choice for NASA, but I don't know how applicable it would be to a military career. Robotics and software engineering are pretty definitely applicable to both. Electrical engineering, chemical engineering, and mechanical engineering are all probably applicable to both, but that's more of an educated guess, and one that's at least partly based on 25-year-old knowledge.

Also be aware, if you're not already, that "employment with NASA" can mean either as a contractor or as a government employee. There are pros and cons to both approaches, and I don't know what the current mix is. In the (science-heavy) building I worked in, contractors seemed to be dominant in software and data analysis, but all of the scientists were either civil servants or folks who were there on grants ("soft money"). I don't have any data on civil-servant engineers.

A good way to get a more objective sense of it might be to check various of the NASA centers' job postings and see what they're looking for. (Note that there are upwards of a dozen, maybe more, scattered all of the country; I think they use a central job-listing database, but I haven't checked.) I don't think that's likely to include contractors, though; you'd probably need to do a separate step to identify which are biggest at NASA and then check their sites specifically. On the military side, the easiest approach would simply be to ask a recruiter. (The answer almost certainly differs for the different services.)

Finally, I'd weight the military side more heavily at this point; that's where you're going first. You might find that you're less interested in engineering and/or in working at NASA once you're done, and even if not, NASA may well have changed by then. (Its budget certainly will have!) Also, I understand that military service tends to be well thought of with regard to management skills, and you might find that you prefer engineering management over straight engineering.