Well, after some scouring of the web, it appears I have my solution. The problem was that I was trying to place the FormChangeButton on top of a PictureBox.
This is not allowed. In fact, as I understand it, no two controls can be placed on top of each other with the lower of the two showing through the upper. The only way for a control to be transparent and display what is below it is for the control to be directly on the page.
So, my solution is to place my cell image on the page itself (Form.BackgroundImage=>image.jpg) and then adjust my FormChangeButton to be transparent
FormChangeButton.Appearance.Backcolor => Transparent
FormChangeButton.FlatAppearance.Bordersize => 0
FormChangeButton.FlatAppearance.MouseDownBackColor => Transparent
FormChangeButton.FlatAppearance.MouseOverBackColor => Transparent
FormChangeButton.FlatAppearance.Bordersize => 0
FormChangeButton.FlatStyle => Flat
FormChangeButton.Font.Text => (Blank)