You are trying to echo PHP code. Additionally you are using single and double quotes wrong.
You should just fix the formatting of your echo() functions. Something like this:
echo '<div id="requestsample"><a class="fancybox button" href="#contact_form_pop">Request a Sample</a><div class="fancybox-hidden" style="display: none;"><div class="hentry" id="contact_form_pop" style="width: 600px; height: 500px; overflow-x: hidden">';
echo do_shortcode( '[contact-form-7 id="269" title="Request Sample"]' );
echo'</div></div></div>';