Here is the response I got from emailing the company:
Thank you for reaching out to us at The Simple Greek. When it comes to Gluten, the reason there is little information aside from meats, is because we use very little items that contain it. The Nutritional information online is accurate for the purpose of ingredients.
Gyro Meat and Falafel Contains it, Gyro Chicken is produced in the same plant, alongside items that contain it. Baklava, Baklava Cheesecake, and the Chocolate Chip Cookies all contain it as well. Pitas contain Gluten
To this end all the other products are completely fine. However, the possibility of cross contamination does exist, so when dining please inform the staff at your location that you have this sensitivity and they will be able to take care of you. If it is a severe sensitivity, I would also ask you to inform them you require a glove change as well, so we can reduce the risk to its lowest we can in our stores.