The evil thing that Egaeus does is proposing. He is supposed to marry his cousin Berenice, and she has accepted. However, he has always thought of her as something to be analyzed, not as someone to be loved. He is particularly obsessed with her teeth.
It is evil to do something like that to someone that you do not love, because marriage should be based on love. Moreover, it is deceitful to the other person, and it ruins their chances of finding someone else who can love them.
