There is not significant amounts of iodine in eggs & onions to worry about. You could however try avoiding them a bit to test for allergy.
Is your acne different since you began avoiding sugar/gluten/dairy? Do you eat citrus? My jawline acne was due to a citrus intolerance. And dairy doesn't noticeably cause acne for me. But there's too many ways that it impacts the events involved in acne formation for me to make it a big part of my diet. Especially unfermented dairy.