Neither degree is required for RO/SRO, however either degree will qualify you to be an STA once you are an SRO, it's generally not additional pay, but extra responsibilities. I believe either degree would open the path for Shift Manager and beyond, although it might require a Master's degree. Personally though, I'd recommend the EE degree, as it opens a lot more doors outside of nuclear power (I'd also expect 2+ years, as very few credits transfer from the navy for actual engineering degrees).
As a side note, University of North Dakota has ABET accredited online degrees for Mechanical, Electrical, Chemical, and Civil Engineering, in case you are interested in those.
