In a small bowl, combine ¼ cup of the pesto, the olive oil, and the lemon juice. Pour mixture into a large zip-top plastic storage bag along with the chicken breasts; seal out excess air and turn bag to coat chicken breasts evenly. Marinade in the refrigerator overnight or at least 4 hours.
An hour before grilling, toss the summer squash, zucchini, and sweet pepper in ¼ cup of the pesto to coat evenly; let stand at room temperature 1 hour to marinate.
Remove chicken from marinade; reserve marinade. Grill chicken on the rack of a covered grill directly over medium heat 15 to 18 minutes or until done (165°F), turning and brushing with reserved marinade twice during grilling. Place pesto-coated vegetables in a grill basket and cook alongside the chicken during the last 8 minutes of grilling, stirring occasionally. Discard any remaining marinade.
