Baffour,
I have well over 80 services setup that point to are SQL databases for data. Some secured, some open to the public.
Additional questions I would be asking you are:
Do you have an ArcGIS Server?
Do you have an ArcGIS Server with rest services open to the public?
Did you setup token security on your ArcGIS Server?
Is the rest service secured? If "Yes". Did you add the rest services into AGOL from your ArcGIS Server? If "Yes". Did you save the user account and password to open the layer up in an AGOL web map. (So users do not need to log into ArcGIS Online and then log into the ArcGIS Server rest services to open up the secured data.)
If you added the data directly in from your rest services (and the service is secured on your ArcGIS Server) you will always be prompted with a login screen.
I do recommend you contact ESRI support. Why?? Because this discussion can lead into several other issues you might be having.
Mike