Ah, the issue is with your SSL. You cannot embed media from an insecure location. You'll need to alter the video embed templates to use HTTPS.
ACP > Templates & Style > Templates > *YOUR THEME* > Ungrouped Templates > video_youtube_embed
Change to:
<object type="application/x-shockwave-flash" class="video_embed" style="width: 450px; height: 366px;" data="https://www.youtube.com/v/{$id}"><param name="movie" value="https://www.youtube.com/v/{$id}" /></object>