There are two issues in play here. First, the 1st Amendment rights of everyone on both sides to speak their minds and spend their money as they see fit. Public debate and protest of contentious. political and religious issues is the very thing the 1st Amendment was written to protect.
The Second, and more important in my mind, is the action of Govt with respect to this issues. When an elected official, using the power of the Government to deny a company or a person whose ONLY offense is offensive speech or support thereof the legal exercise of their business within that Official's jurisdiction, then you get into the realm of 1st Amendment violation.
People are cheering because a politician told a company he would deny them their permits and licenses in his town because he doesn't like what they say or support. If he'd told a LGBT book store the same thing, do you think the same people would be cheering?
Do I support Chick-Fil-A? No. Beyond my dislike for their products in general, I have not engaged in commerce with them for years. I dislike their owner's politics and what he does with the profits from that commerce. That is my right. When appropriate, I engage in dialogues with others about my decision. That is also my Right. However I have never supported anyone attempting to take away the Cathy's rights to support their causes. That is their right.
A larger discussion about the organizations the Cathys support and the actions of those organizations on the local, national and global scale needs to happen. What's going on with Chick-Fil-A is a distraction and smoke-screen to the much larger issues of discrimination, criminalization and legislation of a biological function and personal life choice.