Nope. The first amendment deals only with government restrictions on free speech. You have no such rights when dealing with a private company. The SEC and its schools (and the Flames any anyone else) could kick you off of their property for violating their terms of service. If they say no cameras, and you bring one anyway, they are within their rights to tell you to leave.
As much as the NHL gets trashed at times though, their internet media policies are simply amazing compared to other leagues. While MLB and the SEC do everything they can to keep highlights of their games off sites like YouTube, the NHL is smart enough to realize that what we fans upload to YouTube is free advertising for the league. And the league has their own agreements with Google Video.
Just another reason why hockey is better than any other sport.