Preheat oven to 350 degrees. Tear bread into 1/2-inch pieces (to measure about 1 cup). Transfer to a medium bowl; drizzle with oil. Season with salt and pepper; toss. Transfer to a rimmed baking sheet; toast bread in oven until crisp and golden brown, about 13 minutes. Let cool.
Melt butter in a large saucepan over medium heat. Add onion, garlic, squash, and potato; cook, stirring often, until vegetables begin to soften, about 5 minutes. Add 3 cups stock; bring to a boil. Reduce heat; simmer, stirring occasionally, until potato is tender, 20 to 25 minutes.
Let cool slightly. Carefully ladle soup into a blender (work in batches to avoid filling blender more than halfway); puree until smooth. Pour through a fine sieve into a clean large saucepan.
Place pan over medium-low heat. Stir in remaining 1/2 cup stock; while stirring, slowly pour in buttermilk. Heat until warm, about 5 minutes. Divide soup and reserved croutons among bowls, and garnish with chives.