I used to be worried about chipotle because of the cross contact but I've never gotten sick and my gluten detection dog always passes the food except the outside of the container on occasion. I do not eat the cheese (dairy free) or get anything else that they touch with just their gloves so take that into consideration.
Chipotle does not have a gluten free menu but instead has a list of gluten-containing ingredients to avoid. Cross-contamination is a concern, however, so let the person taking your order know that you are gluten free and if you need them to take the proper precautions.
From their website: "If you avoid gluten, don't eat our flour tortillas. If you are highly sensitive to gluten and would like us to change our gloves, just let us know at the start of your order. You should be aware items containing corn, including corn tortillas (chips, soft corn, and crispy corn tortillas) and corn salsa, may have a small amount of gluten from potentially co-mingling with gluten-containing grains in the field."
This location changed gloves when I asked them to. No reaction. You would want cheese from the back because they use their gloves hands to serve the cheese.