You are not overriding the functions, you are hiding it (in VB its called shadowing, i don't think C# uses this term). In C# the new keyword, when used with a function or property, causes the function or property by the same name and with the same signature in the base class to be hidden. The UserControl.Width and UserControl.Height properties still exist in your user control, but are hidden by your new Width and Height properties.
These shadowed, or hidden, properties can still be accessed, however, and the designer is setting the UserControl.Width property instead of the XYZButtonPane.Width property.
I don't know if this is the best way to do what you want to do, but you could just resize pnlBorder in a Resize event handler of your base class.