What Causes Swollen Lymph Nodes And Sulfur Smelling Stools After Eating Eggs?

I have a daughter, 3 years old with Down Syndrome. If she eats scrambled eggs for a few days in a row, her stools smell extremely sulfur-like. I stop giving her eggs for a few months and then reintruduce them into her diet for a few days... the same thing happens. Also, two weeks after getting the MMR immunization, which has some egg element to it, both lymph nodes between her jaw and ears swelled up terribly for a few days. Symptoms went away and returned two more times the following month, but only on one side. Her daycare looked up some possible Methyl something or other. It was a long word, but it said if infants have this condition then do not feed them eggs among other things.... do you have any information about this that can help?