I'm not sure this a first amendment issue so much as a transparency issue, as they say in the video. Nobody is telling them they can't take the pictures, or write the stories. They are simply telling them they must do it from behind this line. I agree that makes their jobs difficult, and they are certainly hiding quite a bit behind that 65 foot no fly zone, but simply put, there's never full disclosure of details. I agree with the half that say you must keep a distance from the booms, because they don't need to be damaged by careless attempts to take a picture. I'm not sure why they have to be kept so far from the wildlife rescue attempts, it seems like the gov't doesn't want any more shock photos to get out so the press can belittle what already small relief attempts are being made.
And as a sort of off topic note on the subject, I went swimming in Long Island sound this weekend (aka between CT and NY, quite the ways from the Gulf) and there was a light film of oil on the surface of the water. It made this all the more real to me.