Oats are gluten free. Many oats sold in the USA are contaminated with gluten because they are processed in the same factory as gluten containing grains. You can buy Certified 100% Gluten Free oats now.http://www.bobsredmill.com/gluten-free-rolled-oats.html
The oats in the burger recipe are not just a binder but a main ingredient and I would not recommending using Xantham gum as a main ingredient in a recipe.
If you wanted, you could try any gluten free grain (buckwheat, corn, millet, quinoa, etc) as the binder but they may not work as well.
If Gluten is the issue, I would recommend using gluten free oats.