Hi Dave,
I think that Events Tracking of GA is the way to go, but I'm not sure about that in context with YouTube.
But to address the "after <body> tag" requirement, you could open up the Default.aspx file and put it there, but I don't recommend it as you will need to remember that after you do an upgrade so you can do it again.
I usually take the approach of ignoring the "straight" after <body> part of the requirement and I try to put the script as high as possible within the page. That usually means adding a simple HTML module to the highest pane on your page. Put the script inside the Module Headers under the module settings.
This is how I would go about it.
Give it a try and let us know.
Cheers,
Aderson