Nobody’s First Amendment’s rights are being violated here. Anyone who tells you otherwise just doesn’t know what they’re talking about. 
Mr. Duck Dynasty Guy has every right in the world to say whatever hateful things he wants to and there is nothing the government can do about it.
He doesn’t have the right to his own TV show.
If you work for someone, they can regulate your behavior when it relates to your job. If your words or actions can make the company look bad, they can do something about it, and for many big-paying jobs (such as TV star), that’s even in your contract.
The First Amendment prohibits the government from infringing on your speech. It doesn’t protect you against all consequences of your speech.
But conservatives are up in arms over this issue. These are the same people who cheered when liberal commentators Alec Baldwin and Martin Bashir were removed from MSNBC for things they said. They happily approved of radio stations removing the Dixie Chicks from their airwaves because of an interview they did. And they loved it when Bill Maher lost his TV show for his comments.
This is different, though. It must be. I’m sure. I mean, otherwise, these people would be gigantic hypocrites, right?
“But all he did was confirm his conservative Christian beliefs,” they counter. (Personally, I can’t recall the part of the Bible where Jesus talked about how happy blacks were living under Jim Crow segregation laws.) Nothing he said insulted these conservatives.
But is that the standard? “As long as you don’t say anything to upset people who already think exactly like you, you’re fine”?
What if he had said “I think we should kill all gays and worship Hitler”? Would they have objected to him being removed then?
This is just a matter of degree, and the conservatives who are complaining weren’t personally upset by this so can’t understand how others could be. (This helps to support to the whole “conservatives can’t feel empathy” theme — “It doesn’t affect or bother me personally so therefore it isn’t a problem.”)
This was a business decision. A&E knew advertisers would start boycotting the show and they made a financial decision to take action. That’s their decision to dodge the Duck.
And the First Amendment has nothing to do with it.