You should either use (or modify) a pre-existing RDBMS driver for MySQL or make your own and label it as a MySQL driver.
The key piece of information that may be missing is the qualified path to your RDBMS home directory. Squirrel needs to be able to find the jar file that contains the JDBC driver for the RDBMS product that you'll be using.
While I haven't set up Squirrel for MySQL, I have set it up for HSQLDB. In the "Extra Class Path" tab of the change driver dialog, I have added and entered the path to my hsqldb.jar file:
C:\App\Dev\Java\DB\hsqldb\2.2.9\lib\hsqldb.jar
This enables the service provider in the JDBC driver package to locate and load the particular JDBC driver for HSQLDB.
Just as a side note, every RDBMS has its own semantics for the connection URL. The one for HSQLDB is unique to it. The one for MySQL (according to the folks at SquirrelSQL) is:
jdbc:mysql://<hostname>[,<failoverhost>][<:3306>]/
<dbname>[?<param1>=<value1>][&<param2>=<value2>]