Unfortunately I cannot speak to the NewerTech enclosure, but in principle it appears to be possible to have SMART information displayed even for external drives if the enclosure's bridge controller chip allows it.
My experience:
I've used two MacPower Pleiades FW800/400/USB/eSATA enclosures for several years and I've now supplemented them with two additional inXtron enclosures. They all work as expected daisy-chained to the iMac.
The interesting thing is that the inXtron enclosures came with a CD with some software on it ("Smart AP") which is not available for download, notably a Dashboard Widget for displaying actual and maximum temperatures, total hours of operation, start/stop cycles, drive model and serial number.
This widget is complete garbage, though – it is a nauseating nightmare of extreme over-design at the expense of even basic usability:
http://www.inxtron.com/information-center/smart-ap-for-external-hard-drives
To make matters even worse, it can only display the above information for the first external drive connected.
BUT: This Widget internally calls a command line utility which does the actual parameter retrieval, and that one can be called from the terminal as well.
So after installing the software from CD and effectively discarding the Widget, I checked out the command line utility in the Terminal and found that this utility actually reported data about ALL my enclosures, both the inXtron and the MacPower ones (there seems to be firmware compatibility across these):
(The '$' is supposed to be Terminal's bash prompt – type in only the rest of the respective line.)
$ /Applications/MacpowerAP/GetSMART 1
29 50 13 8
53 67 645 15829
52 53 90 7105
51 52 18 13
Each line reports the following for one connected enclosure:
[temp] [maxtemp] [cycles] [hours]
The drives are reported first USB, then FireWire drives, each in the order of first connection / mounting.
$ /Applications/MacpowerAP/GetSMART 2
WD-WCA**************
WDC WD30EZRX-00MMMB0
5TE09***************
ST31000333AS-00MMMB0
5TE09***************
ST31000333AS-00MMMB0
WD-WMA**************
WDC WD30EZRS-00J99B0
Each drive gets two lines reported, first the drive's serial number ("***" blocking out my exact ones), then the drive model.
$ /Applications/MacpowerAP/GetSMART 3
29 50 13 8
WD-WCA**************
WDC WD30EZRX-00MMMB0
53 67 645 15829
5TE09***************
ST31000333AS-00MMMB0
52 53 90 7105
5TE09***************
ST31000333AS-00MMMB0
51 52 18 13
WD-WMA**************
WDC WD30EZRS-00J99B0
This combines both of the above.
$ /Applications/MacpowerAP/GetSMART 5
disk4
disk1
disk2
disk3
This reports the mounted disk names (referring not to the volume names but to the internal Unix names as in /dev/[drive]); Numbering is in connection / mounting order, the actual sequence is as always USB drives, then FireWire drives.
$ /Applications/MacpowerAP/GetSMART 6
USB
1394
1394
1394
This reports the connection interface for each drive in the same sequence as all other commands.
$ /Applications/MacpowerAP/GetSMART 7
MPT934DS1.20.0104.00
MPT934DS1.20.0104.00
This might report the firmware revision or something like that.
These are the commands I know about, and they apply only to the "Smart AP" software supplied with current inXtron enclosures, but it may be possible that other brands use it as well.
I have not been able to identify information about the self-test (early(?) failure warning), but it might be contained as well in the data reported above.
As always, your mileage may vary and you should never do such experiments without FIRST making sure you've got backups of your data and that such backup is not connected during experimentation!
Maybe this can help some people getting the information they need; And maybe there are similar capabilities with other enclosures. At the very least it demonstrates that with the proper firmware in the enclosures themselves and some software on the Mac it IS possible to get at least some of the SMART information from the drive in an external FireWire- or USB-connected enclosure.
None of that is fed into the system's "official" SMART mechanism, though, so no luck using standard mechanisms or applications for it. It's strictly a command-line affair (deservedly disregarding the mishappen Widget).
Good luck!