I have customers connected to my RF600 via several RF560VPNs. I deleted the Packet filter rule for one of them and I was still able to ping myself from their network. If the packet filter is not on, shouldn't that prevent them from seeing my network?

Packet filter rules are only controlling TCP/UDP protocols. You will still be able to ping without packet filter rules. However, you will not be able to access the network without packet filter rules. To test the access without packet filter rules, just try to access any of the services that you have setup on the network and then have to remote workstation try to access it.