Sounds to me like your getting bad chicken. Fresh chicken should have no to a very faint aroma.
I cant imagine over boiling the chicken being the cause of making you feel ill but your particular physiology might not tolerate it as well as someone elses, so i agree with other contributors to your question, try poaching it in baerly simmering water or stock and better yet buy your chicken from a reputable butcher rather than a big market. You might even see if there is a kosher butcher in your area, but if so make sure you specify a non brined chicken.
About the onlyl time i "boil" (really simmer) a breast of chicken is to make a green or veggie salad with. Its quick and easy and then i marinate the cooked & cut up chicken in a vinegrette. Julia Childs recipe in her mastering the art of french cooking is very good. I used to cook without salt, just never felt a need for it, but now a days i put the tinest pinch of salt in my vinegretter, that i make with canola oil and apple cider vinigar, garlic, chopped green onions, black pepper. Marinating the cut up chicken in this vinegrette adds a marvolious layer of flavour to the chicken.
Also you might want to look into the cooking technique of "braising."
Here is a recipe you might be able to adapt to your low salt needs
Here are some links you might find intereting.
--- JL
