The people here are very good about knowing what's gluten friendly on their menu and what isn't. The only thing they will guarantee as gluten free is the burger without bun. However, I've eaten the shakes and fries many times and only gotten ill once, so you may do ok if you're only gluten intolerant rather than celiac.
*update* They now have a gluten free bun, but I wouldn't bother with it. It costs more and quickly disintegrates into mush due to how greasy the burger patties are.