I was hoping there's a way to create the query in XAML alone but it seems like ' for single quote (') is not supported in XAML. You can try out the following:
<Grid x:Name="LayoutRoot" Background="White">
<esri:Map x:Name="MyMap">
<esri:Map.Extent>
<esri:Envelope XMin="-178.217598362366" YMin="18.921786345087" XMax="-66.9692709841984" YMax="71.4062353672549">
<esri:Envelope.SpatialReference>
<esri:SpatialReference WKID="4269"/>
</esri:Envelope.SpatialReference>
</esri:Envelope>
</esri:Map.Extent>
<esri:FeatureLayer ID="MyStateLayer" Url="http://sampleserver1.arcgisonline.com/ArcGIS/rest/services/Demographics/ESRI_Census_USA/MapServer/5" OutFields="*"/>
<esri:FeatureLayer ID="MyCountyLayer" Url="http://sampleserver1.arcgisonline.com/ArcGIS/rest/services/Demographics/ESRI_Census_USA/MapServer/3" OutFields="*" />
</esri:Map>
<StackPanel Orientation="Horizontal" VerticalAlignment="Top" HorizontalAlignment="Center">
<ComboBox x:Name="MyStates" ItemsSource="{Binding ElementName=MyMap, Path=Layers[MyStateLayer].Graphics}" SelectionChanged="ComboBox_SelectionChanged">
<ComboBox.ItemTemplate>
<DataTemplate>
<TextBlock Text="{Binding Attributes[STATE_NAME]}"/>
</DataTemplate>
</ComboBox.ItemTemplate>
</ComboBox>
<ComboBox x:Name="MyCounties" ItemsSource="{Binding ElementName=MyMap, Path=Layers[MyCountyLayer].Graphics}">
<ComboBox.ItemTemplate>
<DataTemplate>
<TextBlock Text="{Binding Attributes[CNTY_FIPS]}"/>
</DataTemplate>
</ComboBox.ItemTemplate>
</ComboBox>
<TextBlock Text="{Binding ElementName=MyStates, Path=SelectedItem.Attributes[STATE_NAME], StringFormat='STATE_NAME = "\{0\}"'}"/>
</StackPanel>
</Grid>
private void ComboBox_SelectionChanged(object sender, SelectionChangedEventArgs e)
{
var stateName = ((sender as ComboBox).SelectedItem as Graphic).Attributes["STATE_NAME"] as string;
FeatureLayer countyLayer = this.MyMap.Layers["MyCountyLayer"] as FeatureLayer;
countyLayer.Where = string.Format("STATE_NAME = '{0}'", stateName);
countyLayer.Update();
}
There's more than one way to populate your combo box with county based on state selection. The code above is just one sample. You can also use QueryTask and store results in a GraphicsLayer and still bind to the Attributes the same way as I did when working with the FeatureLayers.