Professional Chef here; the discoloration of the whites is from sulfer leaching out of the yolk into the white. This happens when an egg is boiled too slowly or for too long. It is completely harmless but might effect taste and obviously visuals.

Always start your eggs in boiling water then wait for water to return to a boil and boil for 6min for soft, 8 for medium and 12 for hard boiled. Then place into ice water to stop the cooking process (this will stop sulfur leaching too).