Correct me if I'm wrong, but I've heard that all this started when the Chick-Fil-A owners answered a question by reporters about whether they support the traditional definition of a family or whatnot.
They answered in the affirmative. They didn't say anything specifically about gay marriage, or go on any rants, or anything like that. They just said that this was their view, and everyone pro-LGBT starts rallying in against them (to the extent of the Chicago Mayor denying them the right to open more stores in the city due to their stated views, which is blatantly illegal).
In response to that, all the hateful anti-LGBT people started rallying in their favor.
Meanwhile all Chick-fil-a did was answer a reporter's question and leave it like that.
So if you want to not go to Chick-Fil-A for that, that's within your rights. But unless I'm missing something, Chick-Fil-A hadn't exactly been actively promoting hatred, or even been a symbol of it until everyone reacted to a simple answer to a question during an interview.