What you have there is a pretty rare situation. An older version of FileMaker with an OS not officially supported by any version of FileMaker.
I personally can tell you that I am running FM12AS on Windows 8 and am able to ODBC into a MySQL database just fine (I know, it's not SQL 2010). I created the ODBC connection as a system DSN. I would try creating a connection from another priogram like Excel and seeing if it works there, just to be sure. If it does work, there could be a compatibility issue...
Also: Make sure you have set the ODBC driver up in the correct version of ODBC--32 bit and 64 bit. If you set up a 32 bit driver in 64 bit ODBC, which you can do, it won't work.