You don't have to go vegan. But eating healthy (fruit, veggies that aren't goitrogens, beans, lentils, whole grains, healthy fish, lean meats, nuts and seeds) will supply you with the nutrients your body needs to repair itself. This is not the only factor in getting well, but I believe it is the foundation.
Be warned though, when going from low carb diets to carbs again can make you feel bad initially. This will be short lived, however. This occurs because the low carbing causes you to become glucose intolerant.