As I said, why not try RadioButton
?
If you only want to achieve this "feature", this is too easy - even don't need involve MVVM
<StackPanel HorizontalAlignment="Center" VerticalAlignment="Center">
<StackPanel.Resources>
<SolidColorBrush x:Key="Button.Static.Background" Color="#FFDDDDDD"/>
<SolidColorBrush x:Key="Button.Static.Border" Color="#FF707070"/>
<SolidColorBrush x:Key="Button.MouseOver.Background" Color="#FFBEE6FD"/>
<SolidColorBrush x:Key="Button.MouseOver.Border" Color="#FF3C7FB1"/>
<SolidColorBrush x:Key="Button.Pressed.Background" Color="#FFC4E5F6"/>
<SolidColorBrush x:Key="Button.Pressed.Border" Color="#FF2C628B"/>
<SolidColorBrush x:Key="Button.Disabled.Background" Color="#FFF4F4F4"/>
<SolidColorBrush x:Key="Button.Disabled.Border" Color="#FFADB2B5"/>
<SolidColorBrush x:Key="Button.Disabled.Foreground" Color="#FF838383"/>
<Style x:Key="FocusVisual">
<Setter Property="Control.Template">
<Setter.Value>
<ControlTemplate>
<Rectangle Margin="2" SnapsToDevicePixels="true" Stroke="{DynamicResource {x:Static SystemColors.ControlTextBrushKey}}" StrokeThickness="1" StrokeDashArray="1 2"/>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>
<Style x:Key="BaseButtonStyle" TargetType="{x:Type RadioButton}">
<Style.Triggers>
<Trigger Property="IsChecked" Value="true">
<Setter Property="IsEnabled" Value="False"/>
</Trigger>
</Style.Triggers>
<Setter Property="FocusVisualStyle" Value="{StaticResource FocusVisual}"/>
<Setter Property="Background" Value="{StaticResource Button.Static.Background}"/>
<Setter Property="BorderBrush" Value="{StaticResource Button.Static.Border}"/>
<Setter Property="Foreground" Value="{DynamicResource {x:Static SystemColors.ControlTextBrushKey}}"/>
<Setter Property="BorderThickness" Value="1"/>
<Setter Property="HorizontalContentAlignment" Value="Center"/>
<Setter Property="VerticalContentAlignment" Value="Center"/>
<Setter Property="Padding" Value="1"/>
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="{x:Type RadioButton}">
<Border x:Name="border" BorderBrush="{TemplateBinding BorderBrush}" BorderThickness="{TemplateBinding BorderThickness}" Background="{TemplateBinding Background}" SnapsToDevicePixels="true">
<ContentPresenter x:Name="contentPresenter" Focusable="False" HorizontalAlignment="{TemplateBinding HorizontalContentAlignment}" Margin="{TemplateBinding Padding}" RecognizesAccessKey="True" SnapsToDevicePixels="{TemplateBinding SnapsToDevicePixels}" VerticalAlignment="{TemplateBinding VerticalContentAlignment}"/>
</Border>
<ControlTemplate.Triggers>
<!--<Trigger Property="IsDefaulted" Value="true">
<Setter Property="BorderBrush" TargetName="border" Value="{DynamicResource {x:Static SystemColors.HighlightBrushKey}}"/>
</Trigger>-->
<Trigger Property="IsMouseOver" Value="true">
<Setter Property="Background" TargetName="border" Value="{StaticResource Button.MouseOver.Background}"/>
<Setter Property="BorderBrush" TargetName="border" Value="{StaticResource Button.MouseOver.Border}"/>
</Trigger>
<Trigger Property="IsPressed" Value="true">
<Setter Property="Background" TargetName="border" Value="{StaticResource Button.Pressed.Background}"/>
<Setter Property="BorderBrush" TargetName="border" Value="{StaticResource Button.Pressed.Border}"/>
</Trigger>
<Trigger Property="IsEnabled" Value="false">
<Setter Property="Background" TargetName="border" Value="{StaticResource Button.Disabled.Background}"/>
<Setter Property="BorderBrush" TargetName="border" Value="{StaticResource Button.Disabled.Border}"/>
<Setter Property="TextElement.Foreground" TargetName="contentPresenter" Value="{StaticResource Button.Disabled.Foreground}"/>
</Trigger>
</ControlTemplate.Triggers>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>
</StackPanel.Resources>
<RadioButton Content="Button1" Style="{StaticResource BaseButtonStyle}"/>
<RadioButton Content="Button2" Style="{StaticResource BaseButtonStyle}"/>
<RadioButton Content="Button3" Style="{StaticResource BaseButtonStyle}"/>
<RadioButton Content="Button4" Style="{StaticResource BaseButtonStyle}"/>
</StackPanel>
Preview:

Then you can setting CommandParameter
for each button to tell the backend code which button was pressed.
You don't need to write any complicated logic to implement this feature. Why not try it?