I would recommend using something like SSH tunnels to carry your JDBC connections through the firewall. Set up a tunnel on the DMZ machine on whatever publicly open port your can, and connect the other end of the tunnel to the appropriate port on the DB server.
Then just change your JDBC connection settings to connect to the tunnel machine's public port and it will transparently end up communicating with the database as usual, while passing through the firewall via the accepted port.
If this is an IT policy problem, in that they won't let you directly access the database, then you would need to work out what you are allowed to do and work with that as far as possible. Changing JDBC to another access method is unlikely to be acceptable to the IT policy in this case.
Edit: after reading Jon's answer, he may be right. I was assuming that the issue was the connection between your server/webapp, and the database server. If you were talking about the client creating direct JDBC connections to the database, then yes - firewall or no, this is very bad practice. The client should talk to your server to ask for what it wants, and your server should do the DB queries as required to get the information.