This reply has so many things to try - thank you, April!
For some answers:
OS is windows 7 on both
Office is Office Professional -- both are 2013.
One workstation is 64 bit with 32 bit Office 2013
We went with .NET 4.0 SP25 for 32 bit Office: EPM Addin Support Package 25.msp,
I did see that this workstation had many older versions of .NET installed including 3.5, so we also tried SP20 3.5 (SP20 is the highest version I have for the 32 bit/.NET 3.5 Office/MS .NET Framework machines).
We needed to install Microsoft Visual Studio 2010 into this workstation, but didn't reinstall 2013.
We didn't reinstall any of the .NET Framework versions. Different departments and different desktop support staff so different escalation protocols and ways of dealing with things.
One workstation is 32 bit with 32 bit Office 2013 so we went with .NET 3.5 SP20.
We also checked to ensure appropriate .NET framework was installed and even reinstalled it.
We reinstalled Microsoft Visual Studio 2010 and 2013.
With this department, the desktop support installed a brand new image into a desktop workstation of a staff member who had just left the department, installed EPM into it and had their EPM user test it out with the same result. This was a 64 bit workstation with 64 bit Office. That really leaves me clueless. I didn't test, myself, in this revamped workstation, but rather took their words for it.
Again, we've uninstalled and reinstalled BPC, deleted and recreated connections and completely destroyed and rebuilt the user profiles (as much as we can - me in BPC Admin and with removing/adding from the VPN group).
I don't think UAC was disabled, but it didn't pop-up during installation -- and it was installed as administrator.
In both instances, MS Office 2013 Pro was reinstalled.
I didn't look to see where mscoree.dll is - I'll add that to the list of things to check-on.
I haven't looked at security event logs - that's a good idea!
Thanks for the ideas - it's given us more things to try!