If you are not able to see Board Stages in External Sharing lease check the following:

1. Ensure the External Sharing Site Guest User is assigned the "Precursive OBX External Sharing" Permission Set, and check this Permission Set has access to Read the following:
   1. Precursive Board object and the Stage field
   2. Precursive Board Type and all fields
   3. Precursive Board Process and all fields
   4. Precursive Board Process Stage and all fields
2. Ensure you have configured a Sharing rule for Precursive Board, Precursive Board Type and Precursive Process to share each object with the Site Guest User.
3. Check that the Owner of the Board Type and Board Process is NOT "Precursive OBX" as this user can interfere with Sharing Rules. If it is assigned to "Precursive OBX" then reassign to another user preferable a System Admin.

If the above does not help please contact Precursive Support.