While it can be done, we typically don't embed videos in the ShoutBox because it slows down the Forum.
Quote[html]<iframe width="100%" src="https://www.youtube.com/embed/p3nXku80y2Y" allowfullscreen></iframe>[/html]
All you need to do is swap
p3nXku80y2Y for the correct YouTube code and paste it all into the ShoutBox.
https://www.youtube.com/watch?v=p3nXku80y2Y
To embed YouTube videos in Forum Posts and your Profile, just add the YouTube video page link/URL. The Forum will take care of the rest.