I'm not an expert on this, but I think it just depends on your son. You can try an egg "challenge test". Avoid eggs for about a week and note his skin appearance, then go crazy and eat as much egg as you can take in the course of a day and check his skin appearance over the next few days. This will tell you if he reacts to the egg in your diet. If he does react most likely there is a medium ground where you can eat some eggs and not affect him.
For the most part, food is not transmitted in breastmilk, just the nutrients in their processed form (milk is made from things in the bloodstream, not the digestive tract). But if I remember correctly in a child who is very very sensitive they may break out depending on mom's diet since of course mom's diet affects whats in her bloodstream.
I would also recommend calling a lactation consultant since they are really the only true experts. Most docs are just not as knowledgable about breastfeeding as lactation consultants are. Most likely the hospital you delivered at has one on staff you can call. Make sure for IBCLC-certified. I am a lactation counselor and it is not the same.