All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Eggs are pure and the moist substances that cover them are also pure, so it is not obligatory to wash them as you have come to know. Thus, the spots that are on them come from this moist substance and if one is in doubt whether it is blood or the moist substance, then in principle they are pure and they are not deemed impure unless there is certainty.
In addition to this, a small amount of blood is excusable according to most scholars. Therefore, if you touch the eggs which have those spots with your wet hands, and you doubt about their impurity, then your hands do not become impure, but if you are sure that they are impure, then, in this case, the impurity is transferred to your hands according to many scholars; so you must wash your hands; but you should not get carried away with obsessive whispers and you should not consider something impure except with absolute certainty.
Allaah knows best.