If we truly believe that business institutions are not 'people', then we shouldn't be attacking them for their leader's beliefs. And if you do believe businesses should be treated like coherent organisms, why stab the heart if it's the head that hurts.
If they break the law or there is evidence of systematic actual job discrimination or customer discrimination / abuse, then that's a different story. But what we have here is a misdirected attack on an independently operated chain.
And as for the whole Chick-fil-a supports killings gays argument, which seems to be the strongest argument for all the hateful words on this issue, it again is misdirected and has so many degrees of separation that it's nonsensical.
Chick-fil-a leaders decide to provide some profit to a separate non-profit arm WinShape. WinShape then takes some of its money and donates to the Family Research Council. FRC then takes some of its money and lobbies the US congress not to denounce Uganda's 'kill the gay' bill - no money is actually directly supporting Uganda's bill - it's being used to not support US denouncement of the bill. If you even believe that's what the FRC's goal was in the first place....
That is a lot farther and a lot different than "Chick-fil-a supports killing gays."