libmysqlclient, the JDBC connector, and other libraries to interfacing to MySQL are GPL (GPLv2). Strict reading of the license would show that you need to distribute your source code under the GPL.
There is the FLOSS exemption, which allows any open source license to include libmysqlclient, however this does not apply to you.
Sun/Oracle aggressively license the connector libraries and server components, and in my experience are quite expensive. There are some tricks you can use, such as a query proxy server - simply launch a child process which can transform your own SQL commands to libmysqlclient. You will need to ship the source of the proxy, but its a self contained piece.