After finally getting some time to test this more, the problem seems to stem from my Mac(s) having network connections to two different subnets.
If I have both a wired connection and a wireless connection, which are on different subnets due to a Google WiFi managing wireless and my Linksys managing wired, then I can not access the unRAID NAS which is only on the wired network.
My ReadyNAS AFP shares can be seen on my Mac regardless if I have either wired, wireless, or both active.
When I move the UNRAID to the wireless network, then my Mac could only see the UNRAID when only the wireless was active.
As I have two network ports in the UNRAID, I'm working around the issue for now by having a direct connection to both subnets which seem to avoid the problem.
My ReadyNAS didn't have any such problems, and I don't see anything special setup for the ReadyNAS in my router.
I don't see any setting on the UNRAID system for AFP accept enable or disable.
Can UNRAID not share AFP across a router?