An interesting question.
Of course, I believe that this effort to remove Chick Fil-A from campus is silly. All these students have to do is simply not go there. Without customers, they would be forced to shut down. But of course, these "high-minded" students no doubt realize that a large percentage of the student body probably has no objections to the owner of this chain exercising his right to free speech so instead, the seek to forcibly impose their morality upon the rest of their fellow students - just like a totalitarian regime would do. Also, it probably makes them feel self-important, which a lot of insecure young people have a need for. Chances are, these students have low self-esteem and are probably getting lousy grades to boot.
Yes, its true what you say of students being insecure and needing “causes” to bolster them.
As far as your question goes, No, I don’t imagine FReepers would mind if conservative students petitioned a gay restaurant. But think about it: Social conservatives have been vastly outnumbered on this issue since the early ‘90s. So the playing field is not level.