Just for clarity, FOG is not a backup system (1 for 1). It is an imaging solution ( 1 to many). It may not be the best use case for your goals.
I can suggest if you ARE looking for a backup solution look at Veeam Agent (free). With this software you can backup your computer to a NAS and create a disaster recover boot drive. If your computer fails, you can boot from the boot drive, connect to your NAS and then recover your computer.
What makes Veeam better than FOG, is how it backs up your disks. FOG does a whole disk clone, where Veeam does a file level backup. Meaning that you can recover any individual file from the backup, where with FOG you would have to restore the entire disk just to recover one file. Veeam is really worth a look.
Will FOG run on a Raspberry Pi 2 or 3, yes. I have a FOG-Pi server setup in my home for development work. For a single capture or deploy it works well in a home lab setup.