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 35
Heat oil in a medium nonstick skillet over medium-high heat.
Add onion; saut 5 minutes or until tender.
Add garlic, and saut 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.