In the bowl of a food processor pulse the basil, parsley, lemon juice, vinegar, and garlic until coarsely chopped. With the machine running, slowly add the oil. Transfer to a small glass or stainless steel bowl and season with the salt and pepper.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Lightly brush the squashes, pepper, and bread on all sides with oil and season evenly with salt and pepper. Grill over direct medium heat, with the lid closed as much as possible, until the vegetables are slightly charred and the bread has grill marks, turning once. The vegetables will take 4 to 6 minutes and the bread will take 2 to 4 minutes. Remove from the grill and let cool.
Coat the chicken on both sides with 3 tablespoons of the dressing. Grill, smooth (skin) side down first, over direct medium heat, with the lid closed as much as possible, until the meat is firm to the touch and opaque all the way to the center, 8 to 12 minutes, turning once or twice. Remove from the grill and let rest for 3 to 5 minutes.
Cut the vegetables, bread, and chicken into 1/2-inch cubes and combine in a large bowl with the tomatoes and feta. Add the dressing and toss to coat. Serve immediately.