So after seeing so many issue regarding the "Connection Error" problem with logging in and having the same issue myself I though I would share how I fixed my particular issue. If for no other reason except to document it should it happen to me again!
Simplify3D was also not the only piece of software I was having this issue with. I also could not connect using Fusion 360 or Spotify on the same Win10 machine. Other Win10 systems on the same network were not having these problems, so I knew it was something specific to this machine.
Mind you I tried all the other solutions presented. Disable the firewall, add an exception for the application, remove and reinstall. All of them and nothing worked. I just stumbled across this and it happened to work for me.
In the Simplify3D login box, I had the Network Settings set to 'Use default system settings'. So I eventually went to check what those system settings were.
I went to the Control Panel (the old style one, not the "clean" Windows 10 one. Type 'control panel' in the Start Menu search). The click on 'Internet Options', then the 'Connections' tab. Here click on the 'Lan Settings' button and make sure that all of these boxes are unticked.
**** IMPORTANT!!! ****
If you actually use a proxy server to connect, then this solution is not for you!!
For some reason my 'Use a proxy server' button was ticked but there were no settings in the address or port boxes, and I do not use a proxy to access the internet.
This solved all the problems I was having with Simplify3D (and the others I mentioned as well). I hope that this helps someone else who is having issues with connection errors.
Timm - Simplify3D user since 2016