So first off, it isn't about your server being IPv6 compliant. It could be an IPv4 only supported server. It is about the device using an iPv6 only network, so that even your IPv4 server looks like an IPv6 server from the perspective of the app.
Adobe staff have confirmed that they tested URLLoader for Apple's IPv6 requirement:
However, I don't think I have seen them confirm explicitly the same thing with URLMonitor, and it seems there might be another user having the same issue with URLMonitor:
Might be good to try and get someone at Adobe to confirm that URLMonitor is IPv6 compliant as well.
You might also be able to find an ANE out there that does similar Reachability code.
Do you have a Mac to be able to test your own local IPv6 network and duplicate Apple's failure?
Silly question I know, but you are using a hostname/domain when using URLMonitor in the app, right? and not hard coding an IPv4 address like "x.x.x.x" (as that will certainly fail in an iPv6-only network)