DayBack and iPhone
DayBack Online is optimized for mobile and touch devices, so anything your can see in your desktop browser, you can see on your phone's browser as well. You can even completely administer DayBack from your phone (changing field mapping, editing resources and statuses, etc.)
Can I Synch FileMaker Events With My Phone?
DayBack doesn't sync events from one source (Basecamp, for example) to another (say, Google Calendar). It brings those sources together in one interface so you can edit them side-by-side.
If you're looking to sync so that users outside of Basecamp (or folks without a DayBack license) can see your schedules, checkout sharing; it will change the way you work.