What's a good way to do this? I have a Synology NAS at home but it's behind a NAT except for specific ports for security cameras. I'll confess I have no earthly idea how to tell if it's been hax0red.

Here's a thread on the Synology forums on how to check if your NAS has been infected and how to remove the malware:

http://forum.synology.com/enu/viewtopic.php?f=19&t=80857&p=3...

In April my university reported several Synology boxes being compromised with "bitcoin" miners. I suspect that was part of this pool.