I've had a chance to look into this and it's actually an issue with the MyBB core. When you press the quote button, the format looks like this:
[ quote='{USERNAME}' ...]
If you change this to:
[ quote="{USERNAME}" ...]
Then it works. Annoying, I know. Hopefully I'll be able to get a fix shoehorned into MyBB for 1.8 via GitHub.