I do stand corrected. The regulation is dated as from 1986. I was a quartermaster from the early to mid seventies. Part of our training was in "Honors and Ceremonies" and flag etiquette was a part of that particular training (I must have slept through some of that class). What I should have made clear in my statement was that a US Navy ship does not dip her national colors (flag) to a vessel rending that salute to her. We never flew foriegn national flags when entering foriegn ports either, at least not to the ports I visited. Special honors due to a boarding VIP (or honoring a special foriegn holiday while visiting that country) is quite a different set of circumstances and in addition to figuring the amounts of saluting rounds to be fired, the amount of sideboys, whether to full dress the ship, man the rails and whatnot, you've got to make sure all of the flags are correct. Navy tradition...
We never flew a "crew's choice" flag either nor do I recall seeing one on any other ship during that time. I see them nowadays on occasion when ships return from deployment. Maybe a later reg covers it.
P.S. What flag that we did fly throughout the fleet in 1976 (Bicentennial Year) was a yellow "Don't Tread on Me" Rattlesnake Flag at the jackstaff. That was a replica of a Revolutionary flag. The "olde":
The present:
PLUS THISEQUALS THIS: