Sync is not intended to be used in that type of situation; i.e., a computer that you don't 'control' or have exclusive use of. Even if you would be able to clear your bookmarks when you are finished with using that 'public' computer [like deleting them in batches via the Library window], the bookmarks backup that Firefox automatically creates wouldn't necessarily be deleted, any user who uses that PC after you might be able to restore your bookmarks. And gawd forbid that a user might be saving their Login data (Passwords) with Sync and be spreading that data to a public computer.

Yes, the Refresh feature doesn't delete bookmarks or Passwords; it's not intended to clear that type of data. Refresh wasn't made for that purpose. https://support.mozilla.org/en-US/kb/refresh-firefox-reset-add-ons-and-settings

Over all, I have misgivings about using a "public computer" that would even allow a user to connect to Firefox Sync and then would allow that Sync data to be 'sync'd' to that PC. IMO, those features should be locked down for the sake of security for the entire network that "public computer" is connected to.

Now, it is possible that public computer is set up to automatically reboot when a user finishes using it - like ejecting the access control card or credit card that was inserted to pay for access - and then upon boot-up a scanning process would automatically clear any data that was saved by the previous user.

You would have to delete the places.sqlite database file in the Firefox profile folder and possible JSON backups in the bookmarkbackups folder to remove the bookmarks and history.

You can use this button to go to the current Firefox profile folder:

• Help > Troubleshooting Information > Profile Directory: Show Folder (Linux: Open Directory; Mac: Show in Finder)
• http://kb.mozillazine.org/Profile_folder_-_Firefox