It looks like the answer is six (plus a 7th guy in the cockpit), based on this photo of USS Newport News (SSN 750) getting underway for a deployment earlier this week:
Interestingly, I don't see anyone with an M-16 up there; I guess that requirement's gone away INCONUS.
Update 1117 08 Oct: An observant reader points out that there is, in fact, a gun visible on the bridge: