I am taking hair treatment with minoxidil. Can I use egg hair mask in between?

Sure, there's no reason why you shouldn't. I won't provide you with significant results in terms of your hair's growth, but maybe it will help with texture/shine. Egg definitely won't harm your hair in any way.

An egg hair mask will not help hair growth in any way. To get protein into the hair, it needs to come through the blood stream - meaning it needs to be eaten.