I'd like to comment on what I think the mechanism is that causes increased prevalence of Hashimoto's thyroiditis and thyroid cancer in ME/CFS. It's based on the Glutathione Depletion--Methylation Cycle Block hypothesis for ME/CFS.
It's known that the thyroid gland normally produces hydrogen peroxide as part of the process of making thyroid hormones. Normally the reactions with hydrogen peroxide are caused to occur just outside the thyroid cells, and the interiors of the cells are protected from oxidative stress by glutathione.
According to the GD-MCB hypothesis (this part adopted from the work of Duthoit et al.), glutathione becomes depleted in the thyroid gland, as it does in certain other organs, tissues and cells of the body. The thyroid cells thus lose their protection against oxidative stress. Some of the hydrogen peroxide (being a neutral molecule) is able to diffuse into the cells. Without sufficient antioxidant protection, the resulting reactive oxidative species react with proteins, lipid membranes, and DNA. The reactions with proteins provoke autoimmune reactions, and this constitutes Hashimoto's autoimmune thyroiditis. Reactions with DNA can produce mutations that lead to thyroid cancer.
There have been some cases of Hashimoto's in ME/CFS that have apparently been reversed by methylation treatment, judging from reports that less or no thyroid hormone supplementation was needed after this treatment was begun. This would be consistent with the above hypothesis.