I tested this before responding. I get the error "Specified Database is not found" when calling
start database '\\machine\share\dir\database.db'
but no error when the statement is
start database '\\\\machine\\share\\dir\\database.db'
You will need to use \\ and not \ in that statement when using UNC pathing. If you use a mapped share, you can refer directly to the file without the extra \'s.
start database 'S:\dir\database.db'