I don't have either of the Garmins, but, in theory, you should be able to transfer route logs between the two.
The file type you should be looking for is .GPX, which is widely used to transfer route data between devices and applications. If you can view your tracks in GoogleEarth, then you're half way there, and must be able to communicate with the GPS receiver. How did you get the tracks from the Etrex into GoogleEarth, for instance? Using conventional manual drag & drop, or did GoogleEarth do it for you?
Provided you know where on your computer the files are stored (and can browse to them using File Explorer, for instance), then you should be able to transfer files back down to the GPS receiver.
I note that the 62stc has a microSD card; that might be one other way of transferring files.
As usual with Garmin, the manual is a disgrace, and provides no useful information whatsoever about where to put tracks so they are visible. The best is can do is on pp45-46 that tell you how to connect the USB cable, and that you can copy files between PC and GPS.
One other option might be to use
http://wtp2.appspot.com/wheresthepath.htm which will import your .GPX file, and, if you have the Garmin Communicator software on your PC, will transfer the file to your receiver.
Hopefully, you'll get a more detailed answer from someone with experience of using a Garmin receiver...