I'm not really sure why the Chik-fil-A thing became news. But since I've been required to pay attention to it, here are some notes
1) They really hammed up the Henson-toy promotion pull on PR by claiming it was a safety issue. That's not helping their position in the slightest to do slimy things to former business partners who disagree with them. To be fair, the mayors of Boston or Chicago should not have come out and declared they might ban the opening of restaurants on this basis. Expressing displeasure as a public official speaking as a private citizen might be fine, but passing legal sanction to formally require everyone else to share in their private displeasure is not.
2) I don't actually care if a corporation is run by former Nazis as long as they are complying with the laws we have regarding discrimination. If Chik-fil-a as a rule, or even as an effect, was treating homosexual or progressive customers differently, I'd be far more concerned than with their political advocacy disagreeing with my views. I haven't even seen evidence of any problem where they might be refusing to hire anyone.
2a) This is a different order problem than the Jim Crow South where the laws imposed discrimination. I'm not entirely convinced that laws imposing non-segregation were unnecessary to help resolve that significant equality issue, but I do think a business that treats some people differently by preventing their business as consumers or labour as employees could be in deep trouble economically if other businesses would not be required to do likewise.
3) I think if people want to boycott a business over a political or public stance on some issue, go right ahead. I don't go there (or KFC) anyway so I'm not all that worked up about it. They aren't losing any business if I declare that I am not going there in the future. I'm not impressed that boycotts typically work in this way however. It is different if it's a civil disobedience matter as with sit-ins in the South where people can be arrested for agitating for equal rights as opposed to an uncertain number of people not conducting direct business transactions. What seems to happen instead is that the announcement of a boycott kicks up support instead of equating always to reduced business. Maybe sometimes they'd work but usually nobody cares for long enough. Whole Foods isn't suffering so far as I know a continued backlash from their CEO's HSA backing op-ed 2 years ago. Perhaps this is different because the issue involved is gaining popular support and appears to be a significant issue for many, many people. Regardless, I'm not going to tell people not to try to express their displeasure with corporate or CEO behavior and statements through economics.
4) Wealthy, powerful, or otherwise successful people often present their personal beliefs and views to the public as an effect of their status and platform of access. We aren't required to agree with these as a matter of course and they are permitted to exercise their rights to speech to say what they want to say. What they should not expect is that people will agree with them and be silent if they do not agree. One way to exercise their disagreement is to stop doing business with their companies and businesses (or for other celebrity type figures, stop going to movies or buying their music, etc). For some companies and figures involved this is easier said than done.
As an example of this, I've been far less fond of the Spurs since discovering that their (now) best player (Tony Parker) was hanging around with "his friend" Chris Brown, a popular musician/asshole when he injured his eye recently in a bar fight. I will be less inclined to watch their games now as a consumer of NBA games. As a counter example, Chik-fil-a is a large corporation with franchised ownership, some of which very likely do not share their CEO's vision of so-called traditional marriage. These people will have much greater difficulty extracting themselves economically than say a random employee or customer but may still wish to differ on their views.
4a) Conservatives, or other people defending Cathy/Chik-fil-a in this, should not confuse the meaning of the 1st amendment protections to free speech to include protection from a) media criticism b) requirements that other people should shut up and be quiet if they don't like what he said, c) that people cannot be fired for what they say by a private entity. The government, as was (briefly) suggested in Boston, should not have much to say about it. That's what the 1st amendment means and protects is government restriction and action. It doesn't protect agreement, disunion, media attention, or even private business decisions and speech restrictions. If you say something that is unpopular, be prepared to be criticised for it, to lose friends or associates, possibly alienate family members, etc.
31 minutes ago