Preheat oven to 400 degrees. Mist a rimmed baking sheet with cooking spray.
In a medium skillet, heat oil on medium. Add squash and shallots and cook, stirring occasionally, until tender, 4 to 6 minutes. Transfer to a medium bowl and stir in bread crumbs, goat cheese, sage, pepper and salt. Set aside.
Place ham on a work surface and top each slice with 1 chicken breast and 1/6th of the squash mixture. Roll up and secure with toothpicks. Arrange on a baking sheet and roast for 25 to 30 minutes. Let rest for 5 minutes. Remove toothpicks and serve.