Heat olive oil in a skillet over medium heat; cook and stir onions in the hot oil until softened, about 10 minutes. Stir in squash and garlic; season with salt. Cook squash mixture, stirring occasionally, until squash is slightly tender, about 10 minutes. Add water if mixture gets too dry. Transfer squash mixture to the prepared casserole dish.

Lightly whisk eggs in a bowl; stir in sour cream, Parmesan cheese, basil, and black pepper. Pour egg mixture over squash mixture; top with mozzarella cheese.

Bake in the preheated oven until squash is tender and mozzarella cheese is bubbling, about 30 minutes.