Disclaimer: this poster is not a lawyer.
The NJ FOID is a state
issued ID. It carries no weight and represents no authority in any state other than NJ. You won't have had your right to own firearms revoked if you're denied- the state of NJ will merely have refused to issue you a form of identification. It should have no bearing on your future ability to purchase and own firearms in other states.
One exception- If you are a NJ resident and show an NJ driver's license when you buy a gun at a store in another state, they may ask to see your NJ FOID. If you don't have one, they may refuse the sale. That wouldn't affect you past that sale on that day; they'd only be refusing you to avoid potential liability.
Now... whether or not the state will find some technicality to deny your FOID application, I can't say. I've had difficulty with handgun permit applications for the horrible offense of being unemployed, so it's entirely possible
they'd try to bust your kneecaps over visiting a psychologist. Depending on who's processing the application, I wouldn't put it past them to deny you for having had the flu.
Kidding aside, there is nothing in the law which states you can't be issued an FOID because you've been to a few therapy sessions. My inclination is to believe you'll be just fine. Best of luck on your application.