Audio Recording Technicians: Is it really best to record audio using flat eq headphones?

Record, not necessarily, as you’re not really performing a lot of mixing and eqing right as you record, but when it comes down to the nitty gritty mixing and mastering, flat headphones or speakers are a must.

When you record, or should I assume you really mean, mix down a song?
Either way, I would use a couple of different speakers because you really want the best sound in the widest range of gear. At some of the studios I’ve been in, they use three or four different kind of speakers for the final mixes.
When you are laying down the primaries, you should have some reference cans, though.
Remember, all audio is subjective. That’s why there are audio controls on ALL audio devices.