This is a slightly complex question, and you will find that the advice from mysql will firmly state that you can't use the mysql connector in a closed source project without paying for a commercial license.
The first thing to say is that, if you don't plan to distribute your product (eg. you will host it on a server, and never give the binaries to a customer to deploy/install/run themselves) then the GPL isn't relevant as it only applies to distribution, and you are free to use the GPL connector.
If you do plan to distribute your application, you could "use" the GPL connector so long as you don't rely on it.
eg. develop your application, testing against both mysql and any other database - ms sql server, or anything. Just try to avoid putting in anything specific to mysql.
Then, you can distribute your product just fine - but distribute it without the mysql connector. You can tell your customers to use mysql or ms sql as they see fit. You're then not distributing the GPL component (and nor is your software dependent on the GPL connector to work), so you are not violating any license.
As others have said, to be sure you should take legal advice, from someone familiar with software licensing law in your jurisdiction.