Yes, thyroid conditions absolutely affect one's appearance. Thyroid disorder, either hyper or hypo, changes the skin, hair, muscle, and bone. For some thyroid patients, they may have increased under eye bags with a tired appearance, or eyes which seem to pop out of their head. Generally, most surgeons performing eyelid surgery will evaluate for thyroid condition, in addition to allergies, prior to any cosmetic surgery. It is very difficult to determine, however, how much of one's appearance is specifically due to thyroid condition vs. normal aging, or other factors.
As far as I know, there isn't any medical necessity for surgical treatment of under eye bags or lower eyelid cosmetic surgery. Insurance may cover upper eyelid or eyebrow surgery, if your visual field is affected.
Speak with a surgeon performing eyelid surgery to help determine if cosmetic eyelid surgery is appropriate for you.