Hi Brandon,

WebDialogWindow does not expose public references to buttons. However, its header has internal variables _closeButton, _minButton and _maxButton (their type is $IG.DialogButton). That object has standard ajax method get_element(). It is also possible to find reference to that html element by traversing within DOM of dialog/header. You also may try to use debugger and check all (internal) member variables and functions.
Below is example to acces html element of min-button.

<script type="text/javascript">
function hideButton()
{
 var dialog = $find('<%=WebDialogWindow1.ClientID%>');
 var header = dialog.get_header();
 //debugger;
 if(!header)
  return;
 var minButton = header._minButton;
 if(!minButton)
  return;
 var style = minButton.get_element().style;
 style.display = 'none';
 style.visibility = 'hidden';
}
</script>

<input type="button" value="hideButton" onclick="hideButton()" />
<ig:WebDialogWindow ID="WebDialogWindow1" runat="server" Height="198px"
   Width="450px">
   <Header>
    <MinimizeBox Visible="True"></MinimizeBox>
   </Header>
</ig:WebDialogWindow>