Preheat oven to 450°.
Melt margarine in a medium saucepan over medium heat. Add flour; cook 30 seconds, stirring constantly. Gradually add milk, stirring with a whisk until blended. Stir in salt and nutmeg; cook until thick, stirring constantly (about 3 minutes).
Spread 1/2 cup sauce in bottom of an 8-inch square baking dish coated with cooking spray. Arrange 2 noodles over sauce; top with half of chicken and half of spinach. Sprinkle with 1/4 teaspoon pepper; top with 3/4 cup sauce. Repeat layers, ending with noodles. Spread remaining sauce over noodles. Cover and bake at 450° for 25 minutes or until noodles are tender and sauce is bubbly. Uncover and top with cheese; bake an additional 5 minutes. Let stand 5 minutes.