Having had no end of problems connecting to my 4tb WD MyCloud since upgrading to El Capitan
Have you updated to 10.11.1?
Pulling the MyCloud apart is almost certainly a warranty violation…
An alternative approach may be to connect a USB drive to the port, then access via SSH (using a device other than your Mac, perhaps?), and copying the user shares to the mounted USB drive. You’d need to have a modicum of Linux experience to find the mounted USB drive, and know where the user shares are (actually, they’re just under ‘/shares’; pretty simple, really…). The point of this method is that everything is done by the MyCloud, and doesn’t use the Mac.
