FOPA only applies if the gun is legal to possess at both the start and end of the trip, and only required stops are made (FOPA is not very clear defining what stops are allowed, but gas would be a required stop).
If you drive into NJ and stop or work, then return home it does NOT apply in NJ.
If you are on your way to New Hampshire it applies.
You have to follow ALL the conditions in FOPA to even try to use it as a defense, and that is all it provides in a number of New England states, a defense in court.
You can refuse to answer or tell the truth.
Lying to the police on anything 'official' is illegal, in and of itself.