(minor spoilers)
I just went to Camp Mccarran and had no problems at first. That was until I approached a door on the second floor -- it's the one that leads to the monorail. There were two NCR soldiers guarding the door, and as I approached, one of them said something like 'no civilians allowed'. They never completed the sentence because in mid-sentence they started firing on me and Rex.

I didn't want to ruin my NCR rep so I ran out and told Rex to stay outside, and went back in. Not only did no one attack me this time at the door, but there was no one on guard there cept for an injured soldier, and he just pvssyd -- said nothing about not being allowed, etc, and let me pass.

My question is -- is this a bug? Or does it have something to do with what the King said about Rex?

Rex tends to attack people with hats, as the King said.

I suspected that, except that it looked like I was going to be fired on before Rex actually attacked anyone. Maybe you are right, and it's just a timing glitch.

Haha I posted the exact same question last night. Just stay away from the door man, it's way too much trouble.

I don't know if it's a bug, those same guards would fire on me (I had only human followers)
also until I did the "I Put a Spell on You" quest. Since I don't have Rex, the dog is not a factor.