Preheat the oven to 375 degrees F.
Heat the olive oil in a medium saucepan over medium heat. Add the onions and cook until translucent, 5 minutes. Add the garlic and pepper flakes; cook for another minute. Stir in the flour; cook for 2 to 3 minutes. Add the milk and cook, stirring, until the mixture becomes thick. Add the cheese, stirring until fully melted. Add the nutmeg and parsley, and season with salt and pepper.
Pour a little of the sauce into a 9-by-13-inch baking dish. Cover the bottom of the baking dish with 3 sheets of the lasagna noodles. Pour in more sauce and spread evenly. Mix the yellow squash and zucchini together in a bowl; layer the lasagna with 2 cups of the mixed squash. Add another 3 sheets of the lasagna noodles, followed by more sauce and then another 2 cups of the mixed squash. Repeat with a final layer of noodles, some more sauce and the remaining 1 cup squash. Sprinkle the mozzarella over the top.
Bake until golden brown and bubbly, about 40 minutes.