If you give them a few days notice, you can order their ice cream cakes GF - they will exclude the cookie crumbs. Super delicious!
Not sure if celiac friendly. I think you can get them to clean out the blizzard machine. I order fries, which I believed are fried with other gluten containing items. I also have ordered an M&M blizzard, Reese's Peanut Butter Cup blizzard, a blue raspberry artic rush, and a grilled chicken sandwich minus the bun. I never got sick. Again, not sure about cross contamination.
If you go into Dairy Queen and tell them you are celiac or can't have gluten, they will change their gloves and change the oil in the fryer, etc...There fries are gluten free and their burgers are too. And they will give you any package to read the ingredients on it if you request it.....
Upon talking to the staff, the fryer is not a dedicated fryer. They fry the cheese curds in the same fryer as the fries. Also the buns are occasionally cooked on the same grill as the burgers. They can make the dessert items for you however. They are more than willing to clean the machines for you.