All very interesting! The role and duties of the Air Force "Flight Engineer" on the HH60G sounds more like a "special ops" crew chief with other duties. If the duties also include aircraft powerplant and airframe maintenance, one most likely could apply for the A&P by experience route.
The FAA flight engineer rating has three categories: Turbojet, Turboprop and Recip. I have never seen a flight engineer panel on a helicopter. Is the HH60G equipped with a engineer systems panel to manage electrical, hydraulic, pneumatic, fuel systems, etc of the aircraft? It's possible some Russian helicopters have a flight engineer station but never heard of it.