Can I have Quicktime player download to an external drive?

QuickTime Player can save to any chosen location, as set from the Options menu in the Controller bar.

For best write speeds, the internal disk would probably be the best initial save location, though.

if I can add my two cents,

I have, in the past, encountered issues with longer recordings stopping / not saving unexpectedly whenever I have selected anything other than "desktop" as my "save to" location. this has been especially true when "save to" is set to an external drive. so I now use "desktop" as my "save to" every time and afterwards I move them to my desired location manually. I haven't had an issue since.