I figured out what's going on. For my case, the problem lied within the map service that I was publishing, not the application consuming the map service. Here's how I fixed things. Hopefully it works for anyone else.
After more searching, I found this posting https://community.esri.com/thread/163168 which spoke about how to utilize date fields when using a feature service. I'm not using a feature service, but figured I'd try it.
In ArcMap, I have a feature class related to a table which has been added into ArcMap as a table. I think that is key, that the table exists in ArcMap, because you can edit the properties of the table.
In the Time tab of the related table, I used one date field. Other than setting the Time Field and Time Zone, everything else is default:
I republished the map services, checked both the Flex and ArcGIS online viewers and found that the related popup dates were correct.
Matt