Use this property to set the style of the drop down border. For example, the following code sets the drop down border color to blue:

<eo:ComboBox>
    <eo:ComboBox.DropDownBorderStyle>
        <Style TargetType="Border">
            <Setter Property="BorderBrush" Value="Blue" />
            <Setter Property="BorderThickness" Value="1" />
        </Style>
    </eo:ComboBox.DropDownBorderStyle>
    <eo:ComboBoxItem>Item 1</eo:ComboBoxItem>
</eo:ComboBox>

Note that because the style is applied as a whole, you may need to have multiple setters in your style even if you only want to change a single property. For example, the above sample sets both BorderBrush and BorderThickness. If you only set BorderBrush, then no border will be shown because BorderThickness has a default value 0 when not set.