For the brine, heat ingredients in a saucepot to dissolve sugar and salt then cool completely.
Place turkey in 2.5-gallon plastic bag and add brine. Let rest several hours, preferably overnight.
If cooking on an outdoor grill or smoker, soak wood chips. For a smoker, add chips to charcoal briquettes or make a foil pouch for gas grill. If using a conventional oven, set turkey on a metal rack placed over a rimmed baking sheet and preheat oven to 325ºF. Heat smoker or grill to 300ºF-325ºF.
Remove turkey from the brine and pat dry. Rub skin with oil and season with ground pepper. Smoke or roast 15 minutes per pound.
For the squash, heat oven to 425ºF. Arrange on a baking sheet, spray with cooking spray and roast slices 25-30 minutes until golden.