You can roast the squash and prepare the bechamel and filling (omitting the egg) up to two days before assembling the lasagna; let each cool before refrigerating in separate containers. Stir the egg into the filling just before using.
Squash: Preheat oven to 400 degrees. Arrange squash in a single layer on 2 rimmed baking sheets. Dot with butter and season with salt and pepper. Roast 5 minutes. Flip squash, coating in butter, and roast 10 minutes more. Flip again and roast until tender, about 2 minutes more. Remove from oven. Reduce oven temperature to 375 degrees.
Bechamel: Meanwhile, melt butter in a medium saucepan over medium-high. Add onion, nutmeg, and 2 teaspoons salt. Cook, stirring occasionally, until onion is translucent, about 3 minutes. Stir in mustard, then wine. Boil 1 minute. Stir in flour and cook, stirring constantly, 30 seconds. Gradually whisk in milk and bring to a boil, whisking constantly. Reduce to a simmer; cook 3 minutes. (You should have 4 cups; add more milk if necessary.)
Filling: Stir together ricotta, 1 1/2 cups Gruyere, and Parmesan. Season with salt, pepper, and nutmeg. Stir in egg.
Lasagna: Cook noodles in a large pot of generously salted boiling water, gently stirring occasionally, until almost al dente, about 6 minutes (or according to package instructions). Drain. Rinse under cold water and lay noodles flat on a lightly oiled baking sheet to drain. (Do not let them touch.)
Spread 1 cup bechamel in the bottom of a 9-by-13-inch baking dish. Arrange 4 noodles on top. Dot with one-third of filling. Top with half of squash, then spinach. Spread 1 cup bechamel over spinach. Top with 4 more noodles, one-third of filling, remaining squash, 1 cup bechamel, and layer of remaining 4 noodles. Finish with remaining filling and bechamel. Sprinkle with remaining 1/2 cup Gruyere.
Cover lasagna with parchment-lined foil and bake until heated through and bubbling at edges, 35 to 40 minutes. Turn oven to broil, remove foil, and broil until golden brown in spots and bubbling, 2 to 5 minutes. Let cool 20 minutes before serving.