The statement of "Access denied for user 'username'@'localhost' (using password: YES)", can surface even before a connection is even made to the database server. I am going to guess this is most likely the issue since I am guessing you know the correct password and I will also make the assumption that you know you have the correct permissions to the tables you are trying to access since you stated you have a user that you use to from your website to make db queries.
You didn't really specify where the eclipse instance is but I am going to guess on a workstation that is not on the same network as your server. I would first make sure that you can make a connection to the database server. MySQL by default runs on port tcp/3306. I would just try making a telnet connection to see if it connects, "telnet (db_server_ip) 3306". If the connection is closed than you will need to open ports from you work station to your db server depending upon the amount of firewalls that are in the way.