Melt the butter in a medium Dutch oven or large saucepan over medium heat. Add the bell pepper and cook, stirring occasionally, until it begins to soften, 3 to 4 minutes. Add the yellow squash and scallion whites and cook, stirring occasionally, until the squash begins to soften, 3 to 4 minutes.
Add the garlic, 1/2 teaspoon salt and a few grinds of pepper to the vegetables and stir. Add 2 cups water and the chili beans; bring to a simmer and cook until slightly thickened, about 4 minutes. Add the chicken and corn and simmer until the corn is tender, 2 to 3 minutes. Stir in the basil and lime juice; season with salt and pepper. Top each serving with shredded cheese, sour cream and the scallion greens.