Halliday's egg is explained clearly in the book. It's based on the first videogame Easter egg, placed in Atari's Adventure game by its sole programmer, Warren Robinett. Whoever finds the egg Halliday hid in the OASIS will inherit his fortune. It's the driving force of the whole plot.
Symbolically, the egg reminds us of rebirth, and that's kind of what happens in the story:
One way that Halliday complicates things is by giving Wade access to the Big Red Button, an emergency shut-off that would delete OASIS entirely. He's pretty much giving Wade the ability to smash the egg into thousands of little pieces. That throws a wrench in our neat little symbol.
Sure, the egg symbolizes Halliday's rebirth in Wade. All of Halliday's passions and interests will live on in his protégé. It's kind of like cloning. However, just like a baby isn't an exact replica of his parent, it's inevitable that Wade isn't going to be an exact copy of Halliday. He'll be a little different, as we see by his socializing with Art3mis-turned-Samantha, a real girl, at the end of the novel. That's something Halliday was never able to do.