AppleScript for Dropbox on External HD

To generate a shared link from a private folder you must make a request to Dropbox's servers using authentication with the Dropbox API. See the docs here. I am not sure you can do that from an applescript, maybe you can use curl. But if you want this in an Alfred workflow you can always use another scripting language like php or python, which are more suitable.

To make sure we are talking apples to apples when referring to dropbox.

In the OS the folder under public called "Dropbox" is not the same as the common "Dropbox" web sharing service. The apple convention allows other users to drop files for you to access without granting other users access to other files in your user folder. The apple "dropbox" is located in ~/public/dropbox/ Once you retrieve a file from that folder you would place it in your own file system wherever you want to access it from normally.

If you are using the "Dropbox" web sharing service you would need to install that program and set it up to sync with one of the folders in your home folder outside of the Public folder.

Assuming you are aware of everything I said above the Dropbox program will need to be updated and reconfigured for El-Capitan as new security elements were established to manage access to network resources from programs and services.

I would suggest uninstalling DropBox from your computer (whatever version you have installed) and then installing the updated version configure to the folder you want to sync and reboot your computer. The files should start syncing then. If they don't check the console to see what is occurring to block the access.