Strange, on my machine it doesn't infinitely loop (although it looks to me like it ought to). Why don't you try:

VB Code:
  1. Private Sub Form1_Resize(ByVal sender As Object, _
  2.     ByVal e As System.EventArgs) Handles MyBase.Resize
  3.         If Me.Width <> Me.Height Then
  4.             If Me.Width > Me.Height Then
  5.                 Me.Height = Me.Width
  6.             Else
  7.                 Me.Width = Me.Height
  8.             End If
  9.         End If
  10.     End Sub

This should reduce the number of times the event fires (avoid adjusting width/height if they are already equal)