WPF dynamic layout: how to enforce square proportions (width equals height)?

The currently accepted answer cannot grow AND shrink. I think I found a better approach, by defining a custom style with a rectangle (stretched uniformly).

<Style x:Key="SquareButton" TargetType="Button">
    <Setter Property="Foreground" Value="White"/>
    <Setter Property="Background" Value="#3a3a3a" />
    <Setter Property="BorderBrush" Value="#5a5a5a"/>
    <Setter Property="Template">
        <Setter.Value>
            <ControlTemplate TargetType="{x:Type Button}">

                <Border HorizontalAlignment="Center">
                    <Grid>
                        <Grid.ColumnDefinitions>
                            <ColumnDefinition Width="auto"/>
                        </Grid.ColumnDefinitions>
                        <Rectangle Stretch="Uniform"
                                         Stroke="{TemplateBinding BorderBrush}"
                                         StrokeThickness="4"
                                         Margin="0" 
                                         Fill="{TemplateBinding Background}" 
                                         Panel.ZIndex="1"
                                         />
                        <ContentPresenter Margin="10" 
                           HorizontalAlignment="Center" 
                           VerticalAlignment="Center" 
                           Panel.ZIndex="2" />
                    </Grid>
                    </Border>

            </ControlTemplate>
        </Setter.Value>
    </Setter>
    <Style.Triggers>
        <Trigger Property="IsMouseOver" Value="True">
            <Setter Property="Background" Value="#4a4a4a"/>
        </Trigger>
    </Style.Triggers>
</Style>

Btw, you can also create a perfect round button this way. Just replace Rectangle with Ellipse.

@Dan Pruzey's answer was good, but did not shrink the square if the window was downsized. What I found to work best was to set the height to auto and the width to that height:

<Button Height="Auto" Width="{Binding ActualHeight, RelativeSource={RelativeSource Self}}"/>

This kept the button a square when sizing my window bigger and smaller. Hope this helps someone.

I think you want to bind button's width to its height, like this:

<Button Name="myButton"
Width="{Binding ElementName=myButton, Path=Height}"
Height="100">
Button Text
</Button>

Try this - it seems to work in Kaxaml:

<Button 
    MinWidth="{Binding ActualHeight, RelativeSource={RelativeSource Self}}" 
    MinHeight="{Binding ActualWidth, RelativeSource={RelativeSource Self}}">
  Some content
</Button>

(To test, I put a TextBox inside the button, because that's an easy way to change content size without re-parsing the Xaml.)

Edit: sorry, should probably have specified it as a style to match your example:

<Style TargetType="Button" x:Key="SquareButton">
  <Setter Property="MinWidth" Value="{Binding ActualHeight, RelativeSource={RelativeSource Self}}" />
  <Setter Property="MinHeight" Value="{Binding ActualWidth, RelativeSource={RelativeSource Self}}" />
</Style>