Preheat oven to 325 degrees F. Spread wheat bread cubes and brown bread cubes in 2 shallow baking pans. Melt 4 tablespoons of the butter and drizzle evenly over bread cubes; toss to coat. Bake, uncovered, for 15 to 20 minutes or until lightly toasted, stirring once. Transfer to a very large mixing bowl.
In a very large skillet melt remaining 2 tablespoons butter over medium heat. Add squash. Cook, uncovered, 5 minutes, stirring occasionally. Add celery and onion; cover and cook 10 minutes more. Stir in 1/4 cup of the chicken broth. Cover and cook 5 minutes more or until squash is just tender, stirring occasionally. Remove from heat. Stir in sage, parsley, nutmeg, salt, and pepper.
Add squash mixture to bread in bowl. Add 2 cups of the remaining broth; toss lightly to coat. Add enough additional broth to make a stuffing of desired moistness.
Transfer stuffing mixture to a greased 3-quart casserole. Bake, covered, for 30 minutes. Uncover and bake 15 to 20 minutes more or until stuffing is heated through. Top with fresh sage sprigs, if desired.