If you are applying for an engineering job without actual experience as an engineer, it may prove difficult to get hired.
I know that the folks we (VC Summer, new units) have hired for engineering have either been experienced engineers or been graduates of traditional (mechanical, electrical, chemical, etc.) engineering programs. Many of them have also interned at operational plants.
Now, having said that it doesn't mean you wont get offered a job. It means that things are stacked slightly against you. Also, what we did on active duty does not count as engineering experience, just as an fyi.
Hearsay alert:
Duke is in the process of closing Crystal River and is looking to relocate those workers to other plants.
TVA has been affected by the government's cuts and has few job postings.
SCANA has been actively recruiting for ops, not sure about engineers, but I would think they'd need more of them as well.
If you have a legit Nuke degree and not some engineering technology degree, then coupled with the Navy background, you would seem a solid choice on paper. Make sure your resume is concise and reflects what they're looking for. There are a couple sites you can use to get in touch with HR types and hiring managers. Be persistent and go get it.
To add to this, Kewaunee will be shutting down in May. Over the next several years those folks will be relocating.