Show button only for certain Woocommerce Product Categories

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>';