Why do you think it's stupid to refuse to give money to a business that actively works against a cause you believe in?
I guess I should rephrase.
If you're doing it because it simply has ruined your enjoyment/usefullness with the act of eating Chick-Fil-A or because you feel good for doing it, and have no issue with understanding this feeling is due to your blissful ignorance, then no it's not dumb.
If you think it's likely to make a significant difference and that's why you're doing it, it's probably stupid. If you're doing it because you think you're taking some kind of principled stand, it's probably stupid. If you're doing it to be able to yap about it on facebook or online, it's probably stupid. And if you don't actually eat at Chick-Fil-A but are proudly proclaiming your boycott of them, yeah that's probably stupid as well.
It's stupid to think it's going to make a significant difference because based on it's geographical location, the general political persuation of people there, and the fact that the amount of people who actually get whipped up in some kind of activist political frenzy pales in comparison to the average population typically, the notion that you're going to harm Chick-Fil-A enough to make it change it's practices in the long run is an extremely unlikely one. It reminds me a lot of voting for a 3rd party candidate because you think it'll make a difference.
It's probably stupid if you're doing it because you think you're taking a principled stand, because the vast majority of people are likely unknowingly hugely inconsistent and are such because of their own chosen and purposeful ignorance and their desire to care about their principles only in so much that it requires them to pay attention to the facebook movement of the week. For those that have spouted off about how it's a voilation of peoples rights or even human rights, do you happen to use an iPhone? HP printer? XBox360? If so then congratulations, you're supporting Foxconn...a company whose human rights record is attrocious. One so bad that you had 100+ employees threaten a mass SUICIDE...yeah, not strike, just flat out killing themselves...if they didn't get their promised raise to around $475 a month (note, month, not week). Or perhaps they're wearing clothes or shoes put together by child labor overseas in horrendous conditions. Or if you want to bring it back home...some of the complaints on Cathy is not HIS actoins but the actions of people he gives moeny to. Is there some mystical end point in which it's no longer a problem? If he gives money to a group that advocates for something is that bad, but if a company you suppot gives money to a group that supports a certain politician/political party that is likely to vote for something you feel is bad (but is not the reason the money was given) is that okay? Because in the end it's the same result...your money going to fund a cause you don't believe in. So to me, taking a hollow principled stand that is principled only if one remains blissfully and purposefully ignorant of everything else they spend their money on, is kind of dumb.
If you're doing it simply to be involved, that should be evident.
And if you're doing it when you weren't even going to eat there anyways, then frankly you're looking for attention even if it's just the attention of the fact you're giving the movement your support.
This isn't even about Chick-Fil-A, but more just in general. If you are forgoing doing something that is beneficial or that you like simply for political reasons when said entity isn't political in nature, where said refusal is likely to have no affect, and where their actions are likely not much different than many things you still participate in with the only exception being the blatantness of it....then to me that's kind of dumb.