Well, there's still guys who are winged Pilots/FO's, then return to the Fleet as Flight Docs. I know it's happened that these folks have got re-stan'ed (assuming they went to an a/c of their old type) and flew as regular mission crew. I don't know whether it's totally kosher under NATOPS, but as I say, I know it's happened. I can't see why it'd be a problem with FOs, or pilots in multi-pilot a/c...don't know about Hornets/Rhinos. Unless it's absolutely 100% blessed by the type wing or CAG, I can't see many skippers letting that one go.
As for the original question - the Docs "assigned" to tailhook squadrons are really part of the Air Wing. Ashore they'll be at the clinic, and underway they'll be down in Sick Bay. They might wear your squadron patch, and you'll call them if you've got a funny itch after a port call, but that's about it. You'll have a corpsman or two assigned to look after your records and make sure your flight physical is up-to-date, etc. More medical yeomen than Docs, at least ashore.
So them coming to fly with you is more of a perk to their job, and a bribe to make sure you can ask them "off the record" medical questions (aka, "Doc, this friend of mine..." questions) should the need arise.