Heat grill or grill pan to medium heat. Grease grill or grill pan with a high smoke point oil.
Grill the bell pepper for 5 minutes on each side, eggplant for 3-4 minutes on each side and zucchini and squash for 2 minutes on each side. Remove the vegetables from the grill and season with salt and black pepper.
Heat a panini press (or large pan) to medium-high heat. Slice ciabatta in half and spread basil pesto on one side and sun-dried tomato pesto on the other.
To the bottom half of the bread, add slices of mozzarella, followed by the eggplant, bell pepper, zucchini and squash. Top with red onion and fresh arugula and close the sandwich up.
Place sandwich in the panini press or large pan. Toast the panini until the mozzarella cheese has melted and the top of the sandwich is golden brown, about 5 minutes in the panini press or 3 minutes on each side in a pan.