Cut eggplants crosswise into 1/2-inch slices; sprinkle cut sides with 1 1/2 teaspoons salt. Place slices in a single layer on paper towels; let stand 1 hour.
Cut zucchini and squash lengthwise into 1/8- to 1/4-inch-thick slices; set aside.
Process 1/2 teaspoon salt, 4 tablespoons olive oil, minced garlic, chopped basil, chopped mint, and balsamic vinegar in a food processor, stopping to scrape down sides.
Rinse eggplant slices with water, and pat dry. Brush slices with olive oil mixture, sprinkle with 1/4 teaspoon salt. Arrange eggplant in a single layer on a lightly greased grill.
Grill eggplant slices, covered with grill lid, over medium-high heat (350° to 400°) 10 to 12 minutes or until lightly browned, turning and brushing with olive oil mixture. Remove eggplant from grill.
Sprinkle zucchini and squash slices with remaining 1/4 teaspoon salt, brush with olive oil mixture. Arrange slices in a single layer on a lightly greased grill.
Grill zucchini and squash slices, covered with grill lid, over medium-high heat 5 minutes; turn and grill 2 more minutes.
Arrange grilled eggplant, zucchini, and squash in a even layer in a 13- x 9-inch baking dish. Pour remaining olive oil mixture over vegetables. Cover and chill 8 hours. Garnish, if desired.