Heat oven to 350° F. Melt the butter in a large pot over medium heat. Add the flour and cook, whisking, until foamy, 30 seconds. Whisk in the milk, ½ teaspoon salt, and ¼ teaspoon pepper. Bring to a boil. Reduce heat and simmer, stirring occasionally, until thickened, 8 to 10 minutes.
Spread a third of the milk mixture in the bottom an 8-by-8-inch baking dish. Top with a third of the noodles, half the kale, and half the squash. Repeat, ending with the remaining lasagna noodles and milk mixture. Top with the Gruyère.
Cover the dish with buttered foil and bake until the filling is bubbling and the noodles are tender, 55 to 60 minutes. Uncover and broil until the top is golden brown, 2 to 4 minutes more. Let rest for 10 minutes before serving.