So, just to make sure I have my thoughts in order on what is, apparently, the burning issue of our times (or, at the very least, a major topic of online discussion this week)...
1. Chick-Fil-A as a corporation supports (through large monetary donations) organizations such as the Family Research Council, which actively campaign against equal rights for gays, lesbians, and QUILTBAG folks in general. (Also, I suspect, against anyone who could be used to scare the people on their mailing lists into donating more money, but that is merely uncharitable speculation on my part.) That is to say, Chick-Fil-A is actively supporting groups who not only don't support equal rights, but actively campaign against them (and in the process, throw out a lot of unfounded accusations and outright lies about QUILTBAG people). This is reprehensible.
2. Since this became national news, a great many people have decided - singly or collectively - that they can no longer, in good conscience, give their money to Chick-Fil-A. This may take the form of actual boycotts, or it may just be a individual decision not to eat there. In either case, this is a valid and appropriate use of free speech and free association rights.
2a. However, it's also worth noting that individual Chick-Fil-A stores are not run by the corporate decision-makers who are primarily responsible for making these ignoble decisions. They are run by franchise owners, some of whom may agree with the corporate stance, and others of whom may have just wanted to make a living running a well-known and popular restaurant in a chain that would guarantee them at least one day off each week.
2b. The employees are even less culpable than the franchise owners, not to mention less likely to actively support this sort of corporate policy. There's no reason to assume that they support these policies, and they have no say in corporate decisions in any case. They don't deserve to be targets for people's frustrations with the way the corporation conducts itself.
3. Several people in public office (including Boston Mayor Thomas Marino) vowed to prevent Chick-Fil-A from opening stores in their areas. This is not okay. This is... I'm not sure "censorship" is exactly the right word, but it's an abuse of government power. Speaking out as a private individual is fine; speaking out as a public official, and using the power of your office to block businesses just because you disagree with the (perfectly legal) views and activities of the people running them, is not. (I've heard some people suggest that these stories are being exaggerated by the media, which is certainly possible; but the Boston Herald quotes the mayor as saying, "If they need licenses in the city, it will be very difficult — unless they open up their policies." So either he was badly misquoted, or he's misusing his authority.) The ACLU has come to Chick-Fil-A's defense on this issue, which isn't terribly surprising if you know anything about what the ACLU actually does.
4. Other people, individually or collectively, have made a point of eating at Chick-Fil-A to show their support for the chain, its policies, or both. This is also a valid and appropriate use of their free speech and free association rights.
5. Speaking for myself, I will not be eating at Chick-Fil-A. Unfortunately, that does mean that some local franchise owners may be unfairly caught in the crossfire; but it's also the most effective way I can express my displeasure with the corporate office for supporting groups that are actively doing harm to people I know and care about.