Yes, subs carry firearms on board. They are usually locked up in the small arms locker. Subs typically carry .45 caliber pistols, 12-gauge shotguns, and M-16s.
We are in the military, after all, and you never know when some firearms might come in handy, you know?
Seriously, the main purpose for the firearms is for the topside watch when tied up at the pier. We want to prevent unauthorized personnel from coming on board, and would like something to enforce this other than just asking them pretty-please.
I never met Rickover. I did meet his successor, Admiral DeMars when I was a midshipman applying for the program.
The reason he interviewed me is because, when Admiral Rickover (the first Director of Naval Nuclear Propulsion) first started the program, he promised Congress that he would personally interview all officers before they came into his program. This practice has continued to this day.
My interview was not as bad as some of the Rickover interviews I’ve heard, but is was still pretty intense. Before you even meet the admiral, you have to make it through the technical interviews. Each interviewer votes you “up” or “down”. You need two “ups” and you get a maximum of three interviews. If you meet a third interviewer, you know you got a least one “down.” (I only had two interviews. 
)
If you make it through the technical interviews, you go before the admiral, who interviews candidates for their ability to think on their feet, potential for leadership, or whatever else floats his boat. Admiral DeMars played mind games with me, but I made it through.
I met him again a few years later when he rode our sub. I asked him if he remembered the interview (when he questioned if I was capable of making it through the program), but he evidently didn’t.