Heat grill to medium. Grill peppers, turning occasionally, until charred, about 18 minutes. Transfer to a bowl, cover with plastic wrap, and let stand for 10 minutes. Rub off skins, and remove ribs and seeds. Slice peppers into 1-inch-thick strips.
Meanwhile, lightly brush zucchini and squashes with oil. Season with salt and pepper. Grill until charred and tender, 2 to 3 minutes per side.
Spread bread slices with pesto. Top 8 slices with vegetables. Season with salt and pepper. Sandwich with remaining bread.
Heat a panini press. Lightly brush sandwiches with oil. Press until bread is golden brown and crusts are crisp, 2 to 4 minutes.