Preheat oven to 375.
Boil lasagna noodles according to package directions. Drain, but keep them in their pot with the lid on to keep them flexible while you work. (No-boil or fresh noodles can be used instead, too.).
Melt the butter in a heavy medium-size saucepan over medium heat. Add the flour and whisk for 1 minute. Gradually whisk in the milk. Bring to a boil over medium-high heat. Reduce the heat to medium and simmer until the sauce thickens slightly, whisking often, about 5 minutes. Season the sauce with nutmeg, salt and pepper to taste, and stir in chopped basil. (A little fresh thyme and/or rosemary is nice here, too.).
Lightly coat a 13 by 9 baking dish with non-stick spray. Spread 1 cup of the sauce over the prepared baking dish. Arrange 1 layer of lasagna noodles on the bottom of the pan. Spread 1/3 of the squash puree over the noodles. Sprinkle with 1/2 cup of mozzarella cheese. Drizzle with another cup of sauce. Repeat layering 3 more times, ending with a layer of noodles. Finish by pouring the remaining sauce over the top and sprinkling lasagna with the remaining mozzarella.
Cover the baking dish with foil and bake the lasagna for 40 minutes at 375. Remove foil and continue baking uncovered until the sauce bubbles and the top is golden, 15 minutes longer. Let the lasagna stand for 15 minutes before serving.