In a 5- to 6-quart pan, combine oil, onion, and garlic. Stir over high heat until onion begins to brown, about 5 minutes.
Add broth and bay leaf. Bring to a boil over high heat.
Cut off and discard peel from squash; cut squash into 1/2-inch cubes and add to pan. Cover and simmer over medium heat until squash is tender when pierced, about 6 minutes.
Meanwhile, trim off discolored ends of Swiss chard stems. Rinse chard, shake off moisture, stack, and cut in half lengthwise; then cut crosswise into 1/4-inch-wide strips.
Add Swiss chard, tomatoes, and beans to squash mixture. Bring to a boil over high heat, stirring often, and boil until chard stems are tender when pierced, 3 to 4 minutes. Ladle into soup bowls.
Season minestrone with salt, pepper, and cheese to taste.