If you gave them a login then you gave them the all clear for all files you handed over the permissions to. Let's remember that good security starts with permissions.

If you didn't hand me the permission to delete, then I can't. If your IT staff didn't lock it down then expect files to go missing.

-> More about this and Event logging. The OS's you listed all have the Event Viewer. Now all you need to add is the events for file deletions. Here's how -> http://infosec.ufl.edu/itsa/nt-jodi.ppt

"My question...is there some kind of login program that could 'silently' record all authorized/unauthorized access to a machine and the users actions so that we could catch who is erasing the security log files?"

If they are in Windows at the time, yes.