Combine water, 1/2 cup oil, vinegar and 3 tablespoons seasoning in a resealable plastic bag. Add chicken. Marinate in refrigerator, at least 30 minutes. Remove chicken from marinade and discard marinade.
Grill chicken over Direct Medium heat for 25 to 30 minutes with the lid closed as much as possible, until the chicken is firm to the touch and no longer pink in the center, with a temperature of at least 170°F, turning once.
While chicken is cooking, slice squashes in half lengthwise. Lightly brush with 1 tablespoon oil. Place on grill and sprinkle with 2 to 3 teaspoons seasoning. Grill 5 to 8 minutes, turning once until squash is lightly browned but still crisp-tender.