Chick-fil-a (hereafter CFA) is a company that makes delicious chicken sandwiches, but they are also run by people who have expressed bigoted views on homosexuality. As a consequence we've seen lots of backlash against the company. People who don't like the views expressed by the company's ownership have organized boycotts and protests, and people who think it's awesome to hold bigoted views on homosexuality (and perhaps also like their chicken) have been patronizing the company as a gesture of political solidarity. This is all business as usual. That said, however, lots of people seem confused about the freedom issues and legal issues that this scenario raises. I'm here to clear that up for you in an uncharacteristically concise manner.
1) Full disclosure: I support marriage equality and gay rights, and I'm disgusted by the remarks of the CFA owners. But I also like their products. And if I stopped buying the stuff of every company whose politics I abhor, I'd be making my own clothing and soap, providing my own energy with a foot-pedaled generator, and firing homemade arrows at pigeons and neighborhood pets for food. Which is why I tend to fight my political battles with my words and actions and not my wallet.
2) No individual or private activist group is violating CFA's free speech by boycotting them or denouncing them. Despite the claims being made by conservatives who have stolen the self-victimization tactic from the left (only without a history of oppression felt by women and minorities to accompany it), there is no First Amendment violation going on here. To say there is suggests a serious misunderstanding of free speech, which is the right to speak your mind, but not to be shielded from criticism and other legal consequences of your speech. The only way this becomes a free speech issue is if city mayors (like Boston's) actually block the company from fair access to zoning licenses; but even there the law suggests that the prospect of having a bigoted company in the neighborhood risks lowering properly value, harming the local economy, etc., such that mayors can actually get away with that kind of thing. The case law allows room for such judgments, though I'm not a big fan of too much leeway here. Nevertheless, the Boston mayor (and others) are well within their rights to speak out against CFA, and this speech in mo way violates CFA owners' own free speech protections.
3) Much of 2 applies regarding freedom of religion as well. CFA is going to get criticized for anti-gay comments, but no one is infringing upon their freedom of religion in making such criticisms. Likewise individuals have the freedom of their own consciences to boycott the company because they don't tolerate bigoted views, religiously based or otherwise. However, if governments do discriminate based on zoning laws, as with the First Amendment discussion above, there may be a freedom of religion case. And there may not. Because...
4) What is actually illegal in many places is discrimination based on sexual orientation. If CFA turns out to discriminate in its hiring policies or it's treatment of employees, then the legal issue becomes a fraught one: not a question of freedom of religion, but of freedom from anti-gay discrimination. And a city most certainly doesn't have to grant zoning licenses to bigoted companies that violate discrimination laws. Mind you, CFA is a a fast food chicken restaurant, not a church or religious organization.