Heat oil in a heavy, large skillet over medium heat. Add the cubed squash and toss to coat. Sprinkle with salt and pepper. Pour the water into the skillet, cover and simmer over medium heat until the squash is tender, about 20 minutes. Transfer the squash to a mixing bowl or food processor and mash. Season the squash purée to taste with more salt and pepper.
Melt the butter in a medium-size saucepan over medium heat. Add flour and whisk for 1 minute. Gradually whisk in the milk. Bring to a low boil over medium- high heat. Reduce the heat to medium and simmer until the sauce thickens slightly, whisking often, about 5 minutes. Add the nutmeg. Season the sauce with salt and pepper to taste.
Preheat the oven to 375 degrees Fahrenheit. Lightly butter a 13-by-9-by-2- inch glass baking dish. Spread 3/4 cup of the sauce over the prepared baking dish. Cover the bottom of pan with one layer of lasagna noodles. Spread half of the squash purée over the noodles. Sprinkle with 1/2 cup of mozzarella cheese. Drizzle 1/2 cup of sauce over the cheese. Repeat layering once more, finishing with a layer of noodles covered only by white sauce.
Tightly cover the baking dish with foil and bake for 40 minutes. Remove cover, sprinkle the remaining mozzarella and Parmesan cheeses over the lasagna and continue baking until the sauce bubbles and the top is golden, about 15 minutes longer. Let the lasagna stand for 15 minutes