No, you don’t have to use Apple Calendar as your primary calendar app. You can use any calendar app on your iOS device, including BusyCal, Calendars 5, Fantastical, Google Calendar, Outlook, Week Calendar and others. The only requirement is that you must configure your iOS device to sync with your calendar services (iCloud, Google Calendar, Exchange, etc.) in Settings > Accounts & Passwords > Accounts.
Once your iOS device is syncing with your calendar services, WhenWorks will be able to access your real-time availability and add appointments to your calendar, regardless of what calendar app you actually use on your device.
A side benefit of integrating with the built-in Calendar on iOS is that you don’t have to create app-specific passwords or deal with other complicated security measures to get up and running quickly. Just install the WhenWorks app, grant it access to your Calendar, and you’re off to the races.