It's really up to you if you take the iodine but if you want proof from the most current research from a man who knows iodine and the thyroid inside out then here is Dr Brownstein's explanation and cure for Hashimoto's which he treats successfully all the time. See page 8 of this link which answers the question "I have Hashimoto's. Can I take iodine?"
The full Iodine Protocol that Dr Brownstein recommends is shown here.