By all means, please boycott Chic-fil-a and mock them mercilessly. But you can’t ban them from cities because of the owner’s rightwing political views. That’s a clear violation of the First Amendment.
Part of what being a liberal is all about is supporting freedom of speech even for your political enemies. Dan Cathy is doing plenty to hurt the anti-gay movement’s standing with moderate voters with this kind of nonsense. Liberals should keep the focus on beating the douchebags who agree with Cathy at the ballot box.