I have Hashimoto's, and while there is a link to gluten intolerance/sensitivity, it is not known to be a causal relationship. In women, most cases are triggered at times of hormonal shifts--e.g., puberty, childbirth, menopause (which is why cases 'peak' at ages 45-65).
With one autoimmune disease, a person is susceptible to others.
Hashimoto's diagnosis via blood tests are notoriously false negative--apparently unless the antibodies are attacking the thyroid at the time of the blood draw, the level will not be high enough for diagnosis. I personally was regularly checked, and my antibody level was low, but when I had a biopsy of one of my thyroid nodules, the pathologist diagnosed Hashi's from the tissue extracted.
Since the treatment is for the subsequent hypothyroidism (as the antibodies destroy the thyroid), the Hashi's diagnosis is not essential--but it's helpful.