MessageBox.Show() displays a dialog. You want to create a dialog as well:
Private Sub picxyes_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles picxyes.Click
Me.DialogResult = DialogResult.OK
End Sub
Private Sub picxno_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles picxno.Click
Me.DialogResult = DialogResult.Cancel
End Sub
Now you can simply use the DialogResult, like you do for MessageBox, in your FormClosing event handler:
If e.CloseReason = CloseReason.UserClosing Then
Using dlg = New ExitGame
If dlg.ShowDialog() = DialogResult.OK Then e.Cancel = True
End Using
End If