I can’t think of any simple way to do this.
There are a number of excellent backup utilities available out there. These can be set to monitor specific folders or even entire drives and any time a change is made to any of the files in the monitored areas, the program will automatically back that data up to wherever you’ve specified.
Some of these programs do this instantly. Others do it once per day. But regardless, you’ll have peace of mind for more than just ZBrush – and you won’t be relying on having to remember to press the “Backup Key”.
Another option is if your new drive is the same size as the original you can set up a RAID 1 array between the two drives. This kind of RAID mirrors the two drives in real-time so everything on one is also on the other. Should one drive ever fail, you simply replace it with another same-sized drive and rebuild the array. This copies all data back over to the replacement drive and you continue as if nothing had happened.