Do the other attendees on a meeting have to be Clockwise users for Clockwise to take their availability into account when rescheduling?

No – due to the way Google Calendar works, Clockwise can tell when the other meeting attendees are free or busy, and only moves meetings into time slots when they are free and within their Google-set working hours.

If the attendees are Clockwise users, then the meeting preferences of that attendee can also be taken into account (when they get into the office, when they eat lunch, etc).