Jefwyn got this right. No one, not the LGBT community, A&E, The media, have told him he can't say what he is saying. They have said that it is not a view they share and they can no longer support him. No first amendment issue. He can continue to say what he wants to say, he just won't have the financial support of A&E.
He has no right to sue A&E. He was fired because he did something that the company found offensive. No different than a person who puts drivetrains in cars saying something derogatory while on the company line.
As for your emails. Yes, you should send them. But what you should really do is get a large group of you together (millions of you) that share the same values and then send a spokesman, just like the LGBT community did. That is how you get heard. Then A&E could weigh out what is important to their bottom line and appease that group. If you could get the head of your church (the big head, not your pastor) to send a message, I am sure A&E would listen.
As for the future of the show, who knows. No, it will never move to another channel. It doesn't mean the clan might not re-appear on another channel ,but Duck Dynasty will be dead as a program. If it does get cancelled, it will have such a stygma no other network will want it. If the clan is smart, they will have the old man apolgize in some form or another (assuming they want the money). If not, the show will go on for one more season without the leader, and then probably fold due to lack of interest.
As for what he said, it was offensive. No one would have been surprised if he has said I don't support gay marriage, or gay rights. But he went that magical step further and put it more in the lines of hate than disagreement.