Okay, so I've also experienced this, however, I did get around it - but the solution is less than ideal for most people. I had gotten a great deal on the Seagate 1.5t drive for $100 at Microcenter, but it started this ejection thing right after I'd re-formatted it for my wifes Mac. Once there, it would dismount mid-time machine, Etc. Terrible. Saw that they had a utility that "fixed" this, and I tried it, and it did not.
The reason it didn't work hadn't dawn on my until after I'd re-cased the original drive in a cheap $21 external drive case (which works perfectly for her).
You see, we had similar, but not identical issues with another soft-powered 1T drive from Toshiba, which would also just shut itself down - but my primary gripe with that was the soft power switch - and you had to keep it near enough to press the button from time to time. When I got this seagate, I chose it because there was no power button - but they decided that they can manage power better than Apple can (a common PC-land mistake when mapped into Mac-Land) - and when that happens, things break. The power management on this drive can be controlled, however, I'd bank that track 0 of the FAT drive (it's original format) likely contains the drive settings - and once I'd reformatted it for MacOS - Seagates ability to read that "control" track was done.
So these drives aren't the blame - it's the firmware on the USB-SATA interface that's lame. I'll bet this firmware reads a track on the drive for it's settings, and that first partition has to be FAT or NTFS or whatever it was for this drive to store, and function correctly.
Folks in MacLand aren't buying this as PC drives and using them with the original PC OS are they? We're all reformatted this to MacOSX - and that step might be the cause of the problem.
I've since re-used a different drive with the old hardware (it is after all a portal 3.5" drive) - and I've not seen the issue yet, but I'm expecting it. If I run into problems, I'll likely reformat it to whatever native format it had and leave it there - then run their utility and test again.
Good luck. I recommend re-casing the drive if you've having issues - USB 2/3 to SATA cases are cheap and plentiful online and at good computer stores like Microcenter. The case I got has a hard power switch, no drivers, and is just a box with a circuit card that the 3.5" drive plugs into - plus four screws to mount the drive and two more to close the case.