Dave,
Frustrating that there isn't a setting in Youtube to automatically play embedded vids in HD (same problem on my own site) but found the following works.
Add the following to the end of your youtube URL.
?rel=0&vq=hd720
Example:-
<iframe src="https://www.youtube.com/embed/xxxxxxxxxxx?rel=0&vq=hd720" frameborder="0" width="640" height="360"></iframe>
Can also use hd1080
Ian.