Preheat oven to 350°. Line a rimmed baking sheet with parchment paper.
In a medium saucepan, bring 2 cups water to a boil over medium-high heat. Stir in wild rice. Reduce heat to medium-low, cover, and simmer until rice is tender and moisture is evaporated, 20 to 25 minutes. In a colander, rinse rice and let drain.
Brush cut side of squash with 2 tablespoons oil. Place squash cut side down on prepared pan.
Bake for 20 minutes. Remove from oven. Turn squash cut side up on prepared baking sheet.
In a medium bowl, combine rice, peppers, apricots, walnuts, preserves, mustard, salt, pepper, thyme, and remaining 2 tablespoons oil. Using a 1-cup scoop, scoop rice mixture into the cavity of each squash.
Bake until squash is tender and rice mixture is warmed through, about 20 minutes.