Is a civilian pilot allowed to fly under part 91 rules with Night Vision Goggles?
Theoretically? Yes. In practice? Maybe. 

The FARs provide for civilian operations under NVGs, but the story is a bit more complicated.

NVG operations requirements for pilots are covered in 61.31(k) for obtaining an endorsement and 61.57(f) for currency. Reading over the wording, the FAA appears to take the same general approach that they do with tailwheels: i.e. get trained and endorsed and then stay current.

The problem is with the aircraft rather, as 91.205(h) requires, in addition to day and night VFR equipment, a gyroscopic pitch and bank indicator (not typically an issue), a generator or alternator as per IFR reqs (again, not generally an issue), interior and exterior lighting appropriate for NVG ops (may be an issue), and a radar altimeter (definitely an issue in almost all light, and some not-so-light, aircraft).

Not sure about the goggles themselves, maybe there's a TSO out there giving tech specifications for aviation-specific NVGs?