Hi, thanks for using health care magic
With a history of persistent cough
and runny nose, it is possible that they have allergic sinusitis
You may want to consider having them tested for allergies to determine the specific triggers, this can be done by blood tests or skin prick testing.
When the specific allergen has been determined then it can be eliminated as much as possible. The use of antihistamines and topical nasal sprays can also be helpful.
The use of the eggs will not affect them unless they are allergic to them.
I hope this helps