Answer
Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
It is permissible to use food and natural substances for purposes other than eating as long as this benefits the person. This is based on the rule, “Permissibility is the basic ruling of things.”
Answering your question, the late Sheikh Ibn Baz, the former mufti of Saudi Arabia, states:
There is nothing wrong with that, if there is some benefit in using henna with egg yolk or any other permissible substances. It is permissible, if there is some benefit to the hair such as making it grow long, conditioning it, etc., or preventing it from falling out.
In addition, the late Sheikh Ibn `Uthaymeen, a prominent Saudi scholar, adds:
It is generally accepted that Allah created food to preserve human life and nourish it. If a person wants to use certain types of food for other lawful purposes like medication, he is permitted to do so.
We add here that seeking to beautify oneself is an Islamic principle because Allah is beautiful and loves beauty. However, a person should avoid extravagance in this and never give it priority over other duties which must be fulfilled.
Allah Almighty knows best.