No. Due to HTTPS encryption, X3watch cannot monitor YouTube videos at this time.
There is now an alternative!
YouTube search and browsing activity can be monitored with Ever Accountable (excluding independent iOS apps).
UPDATE TO EVER ACCOUNTABLE TODAY!
X3watch has partnered with Ever Accountable to bring existing (and new) X3watch customers the next generation of accountability software. Ever Accountable's software works on Mac, Windows, iOS, Android, Kindle Fire tablets and even Chromebooks! Although they do not block websites like X3watch, their accountability reports are second to none. There's so much more information available to users who switch over.
To learn more about making the switch from X3watch to Ever Accountable (and to see a video about making the switch), head over to: https://support.x3watch.com/hc/en-us/articles/115014877028-Existing-X3watch-subscribers-Learn-how-to-set-up-Ever-Accountable