If I give one of them NCR armor, will I instantly be attacked by Caesar's Legion or do they only focus on what the player wears?

Boone would shoot before they do. I don't think anyone else can wear NCR armour.

Most companions will not accept faction armor.
Boone will use NCR armor, Jessica and Arcade Powerarmor, but thats it.
Boone will always attack Legion on sight I think, the rest of the bunch should be neutral as long as you are not attacking anyone.

Companions won't wear (or even carry) faction armour, except Boone who'll wear NCR stuff and Veronica BoS stuff. No-one seems to react to Veronica in BoS gear, and Boone attacks Legion on sight anyway, so it doesn't make a difference with him.

Incedentally, unless it's been changed by a patch, all companions will wear non-faction power armour. I've had Raul running around in Enclave Remnants PA before.

Who's Jessica?

Veronica, me thinks.