OK, I'm making progress, I think.

It seems pretty clear that I do need a designer, and I'm trying to implement
one based on the example in the Help for ControlDesigner class.
Unfortunately, I don't see an event for my designer that relates to placing
the control on the form.  I'm also not getting any event handlers in my
designer to trigger.

I'm obviously missing something.  Here's a code snippet:

<DesignerAttribute(GetType(ShapeControlDesigner))> _
Public Class ShapeControl
Inherits System.Windows.Forms.UserControl
End Class

Public Class ShapeControlDesigner
Inherits System.Windows.Forms.Design.ControlDesigner

Public Sub New()
End Sub

' Sets a value and refreshes the control's display when the
' mouse position enters the area of the control.
Protected Overrides Sub OnMouseEnter()
   ControlSize()
End Sub

Private Sub ControlSize()
   If Not Me.Control.BackgroundImage Is Nothing Then
       MyBase.Control.Size = MyBase.Control.BackgroundImage.Size
       Me.Control.Size = MyBase.Control.BackgroundImage.Size
       Me.Control.Refresh()
   End If
End Sub

End Class