Joe Egg is the nickname Bri and Sheila have given their severely brain damaged child, who is 10 years old at the time of the play. Since she cannot function as a normal human child, they make up conversations for her and invent personalities, though Joe never actually says anything, or even shows any ability to crawl or reach for something.
Her parents make up all kinds of little scenes which they act as if they recounted the history of how Joe got to be so damaged and how many useless therapies and "magics" they had tried to cure her. At one point Bri tries to "let" his daughter die, by not giving her medicine and by exposing her to winter cold, but he doesn't succeed. By the end of the play, he has left Sheila and Joe to themselves.