How do I SSH copy my shares from the MyCloud drive to a USB drive connected on the MyCloud?

Its a similar process explained in your other thread to use rsync to copy from the My Cloud to a USB hard drive attached to the My Cloud. For example:
rsync -rtvh --progress /DataVolume/shares/Public/ /var/media/<USB Share Name>/Public/

Or use a SSH GUI program like WinSCP (for Windows, likely similar programs for Mac) to copy the files from the My Cloud to the USB Share.

Just note that it may not be much faster (if at all) to use a USB drive due to the general slow nature of the USB port used on the My Cloud. One’s copy speed, when direct from My Cloud to another NAS or to a USB hard drive attached to the My Cloud, will also depend on the number of files being transferred and their size. Normally I just issue the rsync command and walk away from the computer for a couple of hours. Unless one uses the nohup command one will have to ensure the computer (or device) that is running the SSH connection doesn’t go to sleep and disconnect or sever the SSH connection to the My Cloud.