The MyBook Essential doesn't have a power switch, as you pointed out. Instead, it takes its power cues from the system it's connected to (i.e. from the USB port). So as long as the USB port gets power, the drive should remain on and functioning.
I just double-checked with my local Apple engineer, and he's pretty sure that no Macs except the latest portables ever power the USB ports off, and they only do it on sleep. Since you're not sleeping the machine, and since you're getting the "improper disconnect" message, I'm about 99% certain your drive is failing to maintain proper communication with the Mac, and is shutting itself off thinking that the Mac is no longer running.
It
could be a faulty USB port. Try the other ports on your machine just to be sure, but since you say it also fails on another computer, I'm pretty sure that's not the problem.
Take the drive back and exchange it for a new one. Better yet, return it, then
look at my answer to this question about good external drives, and buy one of the non-Western-Digital units. I recommend the Mercury Elite drive that's cited last in that answer.
Good luck. Feel free to PM me if you need more help.
