Ordering is done by you on a tablet, so no chance of messing up the order. They have separate entry for gluten and wheat when checking off the allergy list, and when you flag them, a staff would come check your order to make sure that what you ordered are indeed OK with your need. GF buns were delicious and so was the burger great for semi-fast food.
Most Shake Shack locations offer gluten-free buns for burgers. A lettuce wrap is also an option.