Hi there,
If either of you have every done a debloat on your phone, it may have disabled a needed app/service for Samsung Health to login. The item in question is called:
KLMS Agent (com.samsung.klmsagent)
If that has been disabled, it needs to be re-enabled for Samsung Health to work, otherwise you'll end up getting various errors and have issues logging in.
I had the issue where SH would not get past the first screen, and just kept looping. I tried clearing caches (phone and app), reinstalling, installing old versions, etc, but none of that worked. I checked my brothers' Samsung phone, and his worked fine, so I knew it wasn't a SH server issue. I then went into my debloater app, and tried enabling various packages that I had disabled as part of debloating, and after trying a few items, I came across the KLMS Agent app/service, and finally SH got past the first screen and into the app. Once you re-enable that app/service, SH should work.
Hope this helps you out!
