Preheat the oven to 450˚ F. Trim, peel and halve the butternut squash. Scoop out and discard the seeds. Cut the squash into 1/4-inch-thick matchsticks. Coat a rimmed baking sheet with cooking spray. Arrange the squash in an even layer on the baking sheet and coat with cooking spray. Bake until the squash fries are golden and crisp, about 25 minutes. Sprinkle with salt.
Meanwhile, combine the turkey, spinach, scallions, garlic and 1 teaspoon each salt and pepper in a large bowl; mix well. Form into four 4-inch-wide patties. Heat a large skillet over medium-high heat and lightly coat with cooking spray. Add the patties to the skillet and cook until browned and cooked through, about 4 minutes per side, topping each with a slice of Swiss cheese during the last minute of cooking.
Spread the mustard and jam on both halves of the buns. Serve the burgers on the buns and top with lettuce. Serve with the squash fries.