Remove stems and center ribs from Swiss chard. Bring 1/4 cup water to a boil in a large Dutch oven. Add Swiss chard; cover, reduce heat to medium, and cook 5 minutes or until tender. Drain. Plunge Swiss chard into ice water; drain. Squeeze until barely moist; pat dry, and coarsely chop to measure 2 cups.
Combine mushrooms and broth in a large skillet over medium-high heat. Bring to a boil; cover, reduce heat, and cook 3 minutes or until mushrooms are tender. Add Swiss chard; cook 8 minutes or until liquid is absorbed, stirring constantly. Stir in 1/2 teaspoon salt and 1/8 teaspoon pepper; remove from heat.
Combine 1/2 teaspoon salt, 1/8 teaspoon pepper, squash, sage, oil, paprika, and garlic in a large bowl. Spread squash mixture evenly on a jelly-roll pan coated with cooking spray. Bake at 450° for 20 minutes or until squash is tender. Remove from oven.
Reduce oven temperature to 375°.
Combine the squash mixture and the milk in a large bowl. Mash the squash mixture with a potato masher, and fold in Swiss chard mixture. Spoon the mixture into an 11 x 7-inch baking dish coated with cooking spray. Combine the breadcrumbs and melted margarine, and sprinkle over casserole. Cover casserole, and bake at 375° for 15 minutes.