There's a difference here. With the Woolworth's sit ins, the business actually had a policy of discriminating against a group. Here, the owner (or actually a top executive) of the business just expressed a viewpoint. I haven't seen any evidence that discrimination is being practiced.
While there are clearly differences in the specific facts, I see some parallels as well that are noteworthy.
First off, Woolworths was discriminating, but such discrimination was entirely legal. The business was exercising its "rights" consistent with the law at the time. Eliminating discrimination at Woolworths did not involve a change in the law. It was an economic action against an intolerable policy based upon an intolerable point of view of the owners or managers of Woolworths store. Did those idiots have the legal right to discriminate and to hold racist views? Yes. And the remedy was for the public to punish them for their beliefs/policy.
With CFA, the management clearly has the right to hold opinions of hate and indeed to financially support hate groups as they do. But any customer of CFA should know that a portion of what they spend goes to support these hate groups. And the public has the right to make this known, to protest it, and to inflict economic penalties on it.
I would also note that, while many are quick to condemn the mayors for their statements and proclaim that threatening the business or banning it would be illegal, I'm not convinced that is the case. Many cities have adopted anti-hate policies that go beyond Federal limits. My city, for example, bans big box stores, discrimination based on sexual orientation, has banned employees from cooperating in state immigration enforcement, and done numerous other things that I consider stupid. But, those resolutions are the policy of the City until struck down. The mayor and all other city officials are bound by law to implement those policies.
If we had a policy against issuing permits to companies known to support groups that advocate discrimination against homosexuals then CFA would be out of luck until it spent the time and money to challenge the law in court. I concede it would probably win but it would be costly and the discovery process could prove to be embarrassing.
This topic is supposed to be about Chick-fil-a and the so-called appreciation day that some tried to create to whitewash the embarrassment created by the executive's statement. Eagle even tried to further the Republican agenda (which has been well-publicized) to try to cast this as a free-speech issue so that they do not have to confront the marriage-rights issue directly (because public opinion is against them...or at least a minefield). They also want to vilify the mayors because they are all open-minded liberals.
But, this is not a free-speech issue. The owners have the right to say anything and the public has the right to be offended by their views and their use of company revenue to support those views and punish them for it economically with a boycott or other legal actions. To the extent that many cities have passed ordinances opposing the presence of organizations advocating hate, the statements of those mayors might not only be legal, they might even be statements of official city regulations!