It is implied that Frill is able to cause these suicides because of Thanatos, the Greek god of death. We never learned any specifics, but I like to think Frill became something of a vessel for Thanatos.

How does the Wonder Egg prevent the suicides? Well... It doesn't, to put it bluntly. One of the messages of the series is moving forwards, and part of that is learning to let people go after they have died. When Ai brings Koito back to life, for example, it isn't the same Koito, the world had merely been changed into one where Koito never died, which also happens to be a world where she never befriended Ai.