I don't know, but I would assume that these devices would run on battery in the event of power outtages, after all what use would they be otherwise in recording data if the power went down?
Never
assume..
I haven't got the relevant data with me for the 777, but all the other Boeings have these devices on the main elec busses. I can only imagine that this is because the designers think the pilots need the backup electric power more than the recorders. A competely flat battery will increase your risk of crashing considerably.
Would a windmilling fan be able to generate electrical power?
Not at approach speeds. A 747-400 pilot tells me that during cruise, a dead engine remained windmilling fast enough to provide proper power. "N2" or "N3" engine rotor speed (depending on the engine), has to be over 50% or so for the generators to produce power of a quality which will be accepted by the airplane.
Rgds.
NSEU
(Edit) According to the Maintenance Manual D&O, 115Vac power comes from the Right AC Xfr Bus, DC control power comes from the 28Vdc Main Bus. The Left Nose Landing Light uses the Left Xfr Bus and the Left 28VAC Main Bus... but no data as yet for the Right Nose Landing Gear light... but I'm guessing it uses the opposite busses... so the FDR will
probably be getting good power.... but.. we should never assume