How to Make It
Preheat grill to medium-high (400°F to 450°F). Whisk together first 5 ingredients, 3/4 cup of the oil, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper. Place zucchini and squash in a bowl; stir in half of the vinaigrette. (Set aside remaining half.) Let stand 20 minutes.
Brush chicken with remaining 1 tablespoon oil; sprinkle with remaining 3/4 teaspoon salt and 1/2 teaspoon pepper. Grill, covered, turning occasionally, until a thermometer inserted in middle of chicken registers 165°F, 6 minutes. Remove from grill; let rest 5 minutes.
Sprinkle squash mixture with feta and mixed herbs; serve with chicken and remaining vinaigrette.