I figured out how to reset the Youtube video upon hiding thanks to Heathrowe's answer and FigFrance's contribution to the thread found here:
https://forums.adobe.com/message/4979969#4979969
The secret was adding .empty() to my video container. Here is my functioning button code:
Show Video:
//code to embed youtube into symbol
var youtube = $("<iframe/>");
sym.$("video").empty().append(youtube);
youtube.attr('type', 'text/html');
youtube.attr('width', '640');
youtube.attr('height', '360');
youtube.attr('src', 'https://www.youtube.com/embed/MyFv6UKsW70?rel=0');
youtube.attr('frameborder', '1');
youtube.attr('allowfullscreen', '0');
// Show an element
sym.$("video").show();
Hide Video:
sym.$("video").empty();
// Hide an element
sym.$("video").hide();
Brandon