High voltage H-bridge diode
Yes, you definitely need a diode that can withstand the full driver output voltage. The cathode will be boosted to Vcc above that voltage while the anode is held at Vcc.

The lower leakage of a non-Schottky diode will be an advantage, but the slower speed may affect the efficiency of the boost function. However, that part of the circuit usually has plenty of design margin, so it shouldn't be a big deal.