I can't believe we're still discussing this issue.
I've seen this problem since I got my iMac (late 2013 model) last year. Having a bunch of older FW drives connected to my old MacMini, I tried to switch to USB3 on the new iMac I got. Sadly I immediately suffered with the Incorrectly Disconnected messages. I also found that some drives (older Seagates FWIW) also suffered with more serious issues, such as my main LightRoom photo collection would not allow me to write to the drive from LR at random intervals. I also found my Time Machine backup drive became corrupt after a couple of days use, with indications the drive had failed.
Most of my drives are in a four drive enclosure, so none are 'boxed' external drives. It does seem to be worse with Seagate drives though, as other brands seem to work much better. I tried a variety of solutions, from connecting direct to the iMac, and trying a couple of different powered hubs, and changing the cables (despite all being brand new). All to no avail.
In the end I gave up and bought the Thunderbolt to FW800 adapter (luckily the enclosure I bought had multiple connection options), and have had no issues at all, all drives have performed fine since (the only thing is the TB adapter is a little wiggly in the socket, and slight movement can cause a disconnect).
One or two things to try have perhaps come up in this thread, but I'm not sure how much trouble it's worth going too now I have a working solution. I spotted a few things that might relate to my setup, such as the Seagate drives, and using an APC BackUPS power unit, switching off sleep modes, and using Jettison etc. However, none of these seem to offer a genuine fix, nor do they all seem to offer a guarantee to work, and in some cases attempting to fix things could be expensive (replacing drives, hubs, cables and enclosures), but finding the right combination that *might* work is obviously pretty tricky.
There definitely seems to be something screwy with Apple's implementation of USB3 somewhere, and it does extend to more than just sleeping drives too. I found that my iMac was waking up randomly during the night, and mostly within a few seconds of putting it to sleep manually. I eventually found this was coming from a hub that I'd connected using a USB3 cable, although everything plugged in was USB2. Even an empty hub caused the wake up. I changed it to a USB2 cable on the same hub, and now all is well.
My only solution that works satisfactorily was to switch back to FW800 using the Thunderbolt adapter and abandon USB3 altogether.
I would switch to Thunderbolt drives, but the cost of these is so prohibitive to an enthusiastic home user.
Just one final FWIW, I did a bit of searching on this problem in the past, and it's not unique to Macs, but also affects Windows users too, so it would suggest an underlying problem with the USB3 protocol that's been difficult to work around, rather than a muck up by Apple. There have been historic precidences for this too, such as whether to use certain chipsets in PCi cards, or making sure you get Oxford chips in FW devices.