How to use a single URL to log in to Stratus and route the user to private map configurations despite having public access enabled in Spectrum Spatial Analyst?

No, this is not possible. A workaround to this would be to provide a map configuration that only authenticated users have and have them browse to that. They will then always be asked to log in each time.
Environment Details
Products affected: Spectrum Spatial Analyst™
UPDATED:  October 4, 2017