Red-eye flights are typically flights that depart late at night and arrive early in the morning, outside of the common operational day. Since it is a casual term with uses to other types of flights, you will not find a specific indicator regarding flights as red-eye flights.
In the common use for US and Canada flights, red-eye flights are those that depart from the west coast to arrive on the east coast. They would depart sometime between 10pm to 1am and arrive 5am to 7am. When searching, there would appear next to the time of arrival a +1, which refers to arriving the next day (if departure is before midnight the previous day)
Note that when searching for late night options if any flight departs midnight or later it will show on the next day not the day you may be searching for.