Boil the squash until tender. Mash, season with salt and pepper, and set aside. Place the garlic and 2 1/2 cups of the milk in a pot and set over medium heat. When the milk is just barely simmering, place the remaining 1/2 cup of milk and the flour in a bowl and whisk to combine. Whisk this mixture into the simmering milk and cook until the mixture thickens slightly. Remove from the heat; stir in the nutmeg.
Preheat the oven to 350 degrees F. Spoon and spread 1/2 cup of the sauce into a 9" x 13" casserole. Top with 4 noodles and 3/4 cup of the sauce. Spread and top the sauce with 1/2 the mashed squash.
Top the squash with 1/4 of the cheese. Top the cheese with 4 more noodles and another 3/4 cup of the sauce. Top the sauce with the spinach and 1/4 of the remaining cheese. Top with another layer of noodles, 3/4 cup more of the sauce and the rest of the squash. Top the squash with 1/4 more of the cheese and the remaining noodles. Spread the noodles with the remaining sauce and sprinkle with the remaining cheese. Cover the lasagna and bake for 40 minutes. Uncover and bake 10 to 15 minutes more, or until golden brown and bubbly. Let the lasagna rest 10 minutes before serving.
Note: To test this recipe, we used 2lbs. of banana squash. That amount of seeded squash yielded 5 cups of peeled, cubed squash.