What equipment qualifies as FAA-acceptable for providing "Supplemental Oxygen"?
An even more important question than how is how much oxygen to take, but there's no specific FAA guidance or regulation on either point. Different aircraft can carry different equipment, and people have different requirements depending on their age, lungs, blood chemistry, whether they smoke, day vs. night etc. The FAA's guide on oxygen equipment for GA use is extremely limited and although it describes a range of equipment types it doesn't mention any specific requirements.

Having said that, the 'usual' approach seems to be to use a pulse oximeter to measure your oxygen saturation and if it's too low then go on oxygen immediately, regardless of altitude or other considerations. The moderator of AOPA's medical forum is an AME and he recommends oxygen use any time your saturation level drops below 90%, which is apparently also the required level in hospitals for a patient to make an "informed" - i.e. clearheaded - decision.

But returning to the question of equipment, there aren't any regulations on this at all, so provided that the equipment is safe to use in the aircraft (including not obstructing exits and so on) and not illegal under any other law that might apply (carriage of dangerous goods or whatever) then you can go for it. But the usual 'always act as if you'll have to justify your decisions to the FAA tomorrow' advice applies here, IMO, and personally I would always go with an aviation-specific setup if I needed one.