Heat oven to 350° F. Rinse and drain the Swiss chard. Stack the leaves and slice them crosswise into 1-inch-wide strips.
Heat 1 tablespoon of the oil in a large saucepan over medium heat. Add the garlic and cook, stirring, for 1 minute. Add the Swiss chard and squash; stir to coat.
Add the wine, 1 teaspoon salt, and ¼ teaspoon pepper; bring to a boil. Cover, reduce heat, and simmer until the squash is tender, about 15 minutes.
Meanwhile, brush the baguette slices with the remaining oil. Place on foil, sprinkle with the Parmesan, and bake until golden, 15 minutes.
Ladle the soup into bowls and float the toasts on top.