Interesting that you lose weight when not eating eggs. Or maybe it is not weight as such, more like lessening inflammation in the body? It might be that if eggs don’t agree with you and you eat them your body gets inflammed and there is increased water retention. At least I’ve noticed this for myself; I got two sizes bigger and swollen a while ago: either I just got fat in a relatively short time (1 and a half month) while being on the restricted diet (which would be quite depressing!), or then this is water retention caused by inflammation. I am inclined to believe the latter, and have also read that inflammation can cause water retention. So when you lessen inflammation you also lose ‘weight’, possibly?
In any case if you feel better without the eggs you might benefit from taking a break for a few weeks and then trying again after a while. If it is leaky gut related you will be able to eat them again one day when your gut has healed. How long that takes is hard to say. In any case it is best to avoid anything that might cause inflammation or irritate the gut while you are healing it.