Preheat oven to 400°.
Arrange squash in a single layer on a jelly-roll pan coated with cooking spray. Sprinkle with 1/4 teaspoon salt. Bake at 400° for 12 minutes or until tender. Remove from oven; reduce oven temperature to 350°.
Heat oil in a medium nonstick skillet over medium-high heat. Add onion; sauté 5 minutes or until tender. Add garlic, and sauté 1 minute. Remove from heat; cool slightly.
Combine remaining 1/4 teaspoon salt, milk, 1/2 cup cheese, pepper, nutmeg, eggs, and egg whites in a large bowl, stirring with a whisk. Stir in squash and onion mixture. Add bread, and stir gently to combine. Let stand 10 minutes. Spoon into a 2-quart baking dish coated with cooking spray. Sprinkle with remaining 1/2 cup cheese. Bake at 350° for 45 minutes or until pudding is set and lightly browned.