I don't see what the point would be in doing two master's degrees at the same time, but a lot can depend on the details. For example, if you're into two course-based master's degrees that would probably be fine. It would likely take you twice as long to finish that as it would a single degree and so it wouldn't make much of a difference - after about four years you would have two degrees whether you did them in serial or parallel. If you're thinking you can somehow squeeze twice as much work into your time as anyone else and complete such an objective early... well it's not impossible, but you should have a solid, evidence-based reason for believing that you could do something like that.
If you're talking about thesis-based master's programs, that's a different ball of wax. You would likely have a difficult time convincing your supervisor that you should have multiple projects.
And if your final goal is a PhD in nuclear physics, and since you're talking about US programs, it would seem to make more sense to just apply to nuclear physics PhD programs at the completion of your undergrad. If the EE master's is purely for employability purposes, it would make more sense to cross that bridge when you come to it. If you complete a PhD and feel that you need engineering-specific education to get a job doing what you want, that would be the appropriate time to re-train both because it would be based on the current market and in principle give you an up-to-date education.