Position racks in upper and lower thirds of oven; preheat to 450degrees F.

Place chicken in a medium saucepan, add water to cover and bring to a boil. Cover, reduce heat to low and gently simmer until an instant-read thermometer inserted into the thickest part registers 165degrees F, 10 to 15 minutes.

Transfer the chicken to a clean cutting board and shred with 2 forks. Transfer to a large bowl.

Meanwhile, place squash cut-side down in a microwave-safe dish and add 2 tablespoons water. Microwave, uncovered, on High until the flesh is tender, about 10 minutes. (Alternatively, place squash halves cut-side down on a rimmed baking sheet. Bake in a 400 degrees F oven until the squash is tender, 40 to 50 minutes.)

Use a fork to scrape the squash from the shells into the large bowl. Place the shells on a broiler-safe pan. Stir 1 cup enchilada sauce, zucchini, 1/2 teaspoon pepper and 1/4 teaspoon salt into the squash and chicken. Divide the mixture between the shells; top with the remaining 1/4 cup enchilada sauce and cheese.

Bake on the lower rack for 10 minutes. Move to the upper rack, turn the broiler to high and broil, watching carefully, until the cheese starts to brown, about 2 minutes. To serve, cut each shell in half.