try this
<xsl:stylesheet version='1.0' xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
<xsl:output indent="yes"/>
<!-- get keys of order using the date attribute -->
<xsl:key name="dates" match="order" use="@date" />
<xsl:template match="orders">
<xsl:for-each select="order[generate-id()=generate-id(key('dates', @date)[1])]">
<xsl:copy>
<xsl:copy-of select="@*"/>
</xsl:copy>
</xsl:for-each>
</xsl:template>
</xsl:stylesheet>
given your source input, I have the following output
<order date="June-15-2008" item="F101" qty="1" customer="C28901"/>
<order date="June-16-2008" item="F114" qty="1" customer="C28902"/>
<order date="June-17-2008" item="F108" qty="1" customer="C28908"/>