I do recommend performing an elimination style diet to determine if you are sensitive to egg white, however, I would encourage you to do it with the guidance of a coach who can help you meet proper nutrient status while undergoing this.
Just because a product is listed as gluten-free does not automatically make it healthy. Often, when gluten is taken out of a product, it’s just replaced with other cheap ingredients that can (and will) harm your body in other ways.