Place the chayote into a large pot and cover with salted water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, about 15 minutes.
Drain and chop the chayote. Set aside.
Heat the butter in a skillet over medium heat. Stir in the onion; cook and stir until the onion has softened and turned translucent, about 5 minutes. Stir in the chopped chayote, and season with salt and pepper. Break the eggs over the chayote mixture, and reduce heat to low. Cook and stir until the eggs have set, 3 to 4 minutes.
Transfer the mixture to a baking dish and top with American cheese.
Bake in the preheated oven until the cheese has melted, about 5 minutes.
