I checked Snopes and could not find anything on it.
IIRC there is something that requires the VA to report names of those found to be adjudicated as insane, a danger to self or others, or unable to manage their affairs (probably not the precise terms in the law) to NICS (which is not Homeland Security).
The operative word is "adjudicated" - needs to be a formal, adversarial hearing, not merely a panel of three docs saying you need the rubber room.
The "Three Questions" are pretty standard screening for anyone coming in seeking mental health services, but I were asked those because I wanted treatment for arthrits I would be pretty much upset and thinking the nurse might need some rubber room time.
By the way -
Do you know what day it is?
Do you know who is President?
Do you know where you are?
I'm going to say three numbers and three words. Try to remember them. After your waterboarding I'll ask you to repeat the three numbers and three words.
Put the foil beanie away.
stay safe.
skidmark