Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Chicago O'Hare Intl Airport to Hurghada as you normally would.
Due to multiple factors, expect to potentially pay more when booking in June. For low rates, July can be a great choice for travel to Hurghada. Flights to Hurghada from Chicago O'Hare Intl Airport in July are about $979 on average during July, but can be found for as low as $694.