MySQL can be used by SQL Developer provided by Oracle. It is free & can be downloaded from- http://www.oracle.com/technology/products/database/sql_developer/index.html. There is no any default driver for MySQL in Oracle. For this you have to download JDBC connector for MySQL. Download it from- http://www.mysql.com/products/connector/j/
Extract the file & place into a directory you wish.
Now start SQL Developer and click on the Tools>Preferences. It is like the image given below-
Click on the Add Entry button, navigate to the directory of the downloaded connector JAR file & select it.
Select MySQL from the database options & choose the database you want to work with. AND it is done!
But remember, if your user in the MySQL have no password or null, it will not be accepted in SQL developer. Give some damn password & it will work fine.