Having tried to get my head around why I developed an autoimmune disease (psoriasis) at at fifty-one I've come to the conclusion that, in general, autoimmune diseases are initiated by environmental factors and a person have a certain genetic make-up. Environmental factors can include pathogens, exposure to toxins (incl. heavy metals), lifestyle excesses (drink, smoke, poor diet), stress ... or *any combination* of the above. One's genetic make-up will determine which autoimmune disease will be sparked, if any. So in my humble opinion, it is probably impossible to determine why any individual develops an autoimmune disease ... including Hashimoto's. Could Iodine deficiency cause Hashimoto's? Possibly, but it would best viewed as only one of many possible contributing factors.
Also the genetic factor should not be overlooked. Depending on the autoimmune disease oftentimes there is greater than a 50% chance a child will get the disease if a parent has it. In the case of Hashimoto's would Iodine supplementation prevent a child from developing the disease? Hard to say.
If one has Hashimoto's it would be best to focus on doing all the right things to calm one's immune system, diet is especially important, and taking supplements that the thyroid needs. Getting thyroid antibodies tested regularly (every 3 months?) is a must to gauge one's progress.