iodine rich food in hypothyroid patients does not help if the cause of hypothyroidism is autoimmune thyroiditis, in contrary it can trigger the autoimmune process.
A sufficient iodine intake can prevent from iodine deficiency goiter, that's all. Hyperthyroid patients should be prevented from iodine rich food!
Brassica (cruciferous) vegetables contain progoitrin and glucosinolates, which may interfere with iodine uptake by the thyroid gland and thyroid hormone production. Only a few species (e.g. collards, Brussel sprouts, Russian kale spp.) contain higher concentrations, which may potentially have clinically relevant effects (Article Concentrations of thiocyanate and goitrin in human plasma, t...
). However, cooking reduces the goitrogenic effect. Consumed in "normal" amounts, the risk of adverse effects is very low. The same concept applies to isoflavone-containing soy food (
Article Effects of Soy Protein and Soybean Isoflavones on Thyroid Fu...