Yes, children can eat whole eggs as soon as they are introduced to solid foods, regardless of any predisposition to allergies.
For many years, it was recommended to wait until a child was one year old before introducing egg whites; as this was thought to prevent the onset of an egg allergy. However, there is no scientific evidence to suggest that introducing eggs after the age of one reduces the risk of an egg allergy.
On the other hand, certain studies suggest that the later the introduction of an allergenic food, the higher the risk of developing an allergy to that particular food.
This new recommendation applies to all infants, regardless of whether their immediate family members have allergies.