Here's all it takes.
1) edit sqldeveloper/bin/sqldeveloper.con
- add the path to the Dropbox or where ever location
2) Start it up.
The to check just look in that location and you'll find a .sqldeveloper directory where everything will be stored.
You'll notice bunches of directories where all the settings are being stored.
Then just go to any machine with your drop box installed and you'll have the exact same setup everywhere.