I don't know how to tell him it's not his.
Usually, when a couple discusses a child "not being his," it refers to a woman's infidelity, which is a hard topic to discuss. Here, however, it's not a partner cheating, it's a man nesting on a giant egg: the egg obviously isn't his child because humans don't lay eggs, and so the usual expression takes on a humorous new meaning: the woman can't seem to convince the man that the egg isn't his child, despite the fact that it's obvious it couldn't be.