Place a large foil pan on top of the Flavorizer bars, below the cooking grate. Fill it halfway with room temperature water. Prepare the grill for cooking over medium heat (400°F).
Wash the squashes under cold water. Using a heavy, sharp knife, cut the ends off of each squash and then cut each squash into four or six pieces. Using a tablespoon, scoop out and discard the seeds and pulp. Place the squash pieces on a baking sheet and brush the flesh with the oil. Season with salt and pepper.
Brush the cooking grates clean. Remove the squash pieces from the baking sheet. Grill the squash pieces directly on the cooking grates, skin side down, above foil pan, over medium heat, with the lid closed, for about 30 minutes. Meanwhile, make the glaze.
In a small saucepan over medium heat on the stove, combine the glaze ingredients and cook until the butter is melted and the glaze is smooth, about 2 minutes, stirring often. Remove from the heat.
After the first 30 minutes of cooking, return the squash pieces to the sheet pan, skin side down. Close the grill’s lid to maintain the heat inside. Brush the flesh with the glaze, return the squash on the pan to the grill, and continue to roast, above foil pan, over medium heat, with the lid closed, until soft and tender, 30 to 40 minutes more, glazing every 15 to 20 minutes. Season with more salt, if desired. Drizzle with the remaining glaze and serve warm.