OK, just completed a screencast ... and Kazam crashed out on me!
1. OK, download the correct package for system (64-bit or 32-bit for Ubuntu as that is what Zorin is based on) from dropbox site and install using 'gdebi Package Installer'.
2. Whilst drop-box is installing you should see the drop-box icon appear in the panel at the bottom of the screen (AWN panel) - while it is there, right-click and select 'create custom launcher'.
3. Right-click the AWN Panel and select 'Preferences' then select 'Applets Tab' - scroll down to 'Yet Another Menu Applet' and drag this to the bottom panel within 'Applets' - you should see another 'Z' appear - right click on this and select 'Edit Menus' - go to the Internet section and in the right-pane you should see 'Dropbox' - select this menu item then click on the Properties button to see the command line and descriptors.
4. On the create custom launcher that you have open, this should be 'blank' - copy and paste each field into the new custom launcher.
5. To get the applications icon to show instead of a question mark/exclamation mark (poor memory!), click on the icon in the new custom launcher properties dialog and browse to:
/usr/share/icons/zorin light/scalable - then scroll-down til you get to the dropbox.png and select open - voila, you now have a custom launcher!