Do due dates, priorities, and To-dos sync across devices?

To-dos, due dates, and priorities cannot currently sync between devices.

This feature is in the plans though, and you can help to prioritize it by adding your vote to the feature request forums here: 

iOS:
http://boxer.uservoice.com/forums/236855-feature-requests/suggestions/5484804-sync-due-dates-and-priorities-across-devices

Android: 
http://support.getboxer.com/forums/256309-android-feedback/suggestions/6390071-sync-priorities-due-dates-and-assignees-across-d

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