People continuing to talk about Duck Dynasty is like beating a dead horse. Duck Dynasty is a show I do not know anything about and have never watched. Everyone has the right to free speech and if he lived in Kentucky, the Kentucky Constitution protects this right more so than the 1st Amendment to the U.S. Constitution. However, when you speak, consequences sometimes follow, especially if you are on a television show owned by a network such as A&E and you are being PAID by them!
These people are basically employees in some fashion of A&E, the owner of the show, and subject to A&E's mission statement and values.
Both employees and independent contracts are subject to termination or remedial action with or without notice. The End (the curtain comes down and now everyone makes an exit).