>It's my understanding that it's better to put the code in the code behind page. Is that correct?

I haven't been able to think of a reason why one would be any better than the other, especially if you used VS.NET.

>If I add System.Data.SqlClient to the Dim and New statements, everything works fine. Can someone tell my why I have to explicitly reference System.Data.SqlClient?

You don't have any classes in your own Namespace named SqlConnection, do you? If you don't use the System.Data.SqlClient namespace, what does IntelliSense give you for members if the SqlConnection class?