I personally have never tried to add mySQL to Eclipse but I found this guide, in step 2 they install the Connector/J driver so their program can connect to MySQL, you could check it to see if it helps you. This other guide looks even more complete, it might be worth looking at it as well.
Let me know if those documents help you install mySQL connector.
I forgot to mention that I already have the Connector installed and it works fine when creating a regular Java project, however it throws the ClassNotFoundException when running the code on a Java Intel IoT project.
I see, can you share your code or an example code? I would like to test it to see if I can replicate the issue.