This is an excellent question. While the obvious answer would be no, there are caveats.
"Mechanical switch" is an ambiguous term. Let's assume for a moment that a mechanical switch is one that is actuated by a broad movement (as opposed to pressure).
Let's take a look at a typical PSU switch:
See those thick prongs? The electrical current that powers the device actually flows from those, and turning the switch off is very much equivalent to unplugging the cable from the wall power socket.
A pc case power button, on the other hand, looks very different internally:
Those thin wires? They don't actually carry any current (that powers the device). Your computer is, in fact, powered all the time, and the switch just tells it to "turn on". This is also known as "on standby" and "soft-off".
The fact that the power switch moves or stays in place is completely irrelevant. What matters is what the actual circuit connects to - and you can't see that without opening the device.
With that being said, switches like the first one are most commonly powering the devices themselves.
Now, if your hard drive switch uses a "standby" mode, and the controller of the chip has a chip that is writable and you can do it through the USB interface (as opposed to on-board headers, which would be most common) theoretically malware could overwrite the firmware with a special version that turns on/off at will.
To keep this interesting, let's for a moment ignore the fact that if the disk is ever turned on you'll be able to hear it spinning.
That malware would have to be developed specifically for your particular hard drive maker and model (and possibly factory series), so personally I don't think this is a realistic attack. It's much easier to get someone to break the window of the room the disk is in.
Of course, if your disk switch actually turns off the power, this is even more interesting. Malware certainly can't turn it back on, but it can:
Make your computer reboot and fake a operating system message telling you it needs to check the filesystem integrity in the external hard drive, and refusing to boot until you power it on
Access the files through the same mechanism your backup system is using to get them...
Just wait for it!