To hide the Gift Aid section on the Donation form please add the following code before the generated Online Express form embed code when attaching to your page:
 
<style>#bboxdonation_gift_fldUKGiftAid {display:none;}</style>

Example of complete code:
<style>#bboxdonation_gift_fldUKGiftAid {display:none;}</style> 
<div id="bbox-root"></div>
<script type="text/javascript">
       window.bboxInit = function () {
           bbox.showForm('XXXXXXXXXXXXXXXXXXXXX');
       };
       (function () {
           var e = document.createElement('script'); e.async = true;
           e.src = 'https://bbox.blackbaudhosting.com/webforms/bbox-min.js';
           document.getElementsByTagName('head')[0].appendChild(e);
       } ());
</script>