OK, first things first. You will NEVER get a stupid supervisor to agree that anything is wrong with their people until you rub their noses in it. So...
BEFORE you do the closeout stuff behind the scenes, make a log entry to an external file, opened in VBA, for append, showing date/time at which this user started the shutdown. Close the file IMMEDIATELY. THEN make a second entry a couple of minutes later showing that the shutdown is complete. Again, close the file IMMEDIATELY.
Put the shutdown file in a sub-folder on the server where you DB is located so it isn't "in their face" when they open the folder with the DB. AND despite many Access security requirements, it IS permitted that in a sub folder you would deny them DELETE access to some or all files.
I believe there is an advanced permission that would allow a user to INSERT TO but not UPDATE or DELETE FROM a given file. Get your security manager to help. I'm not 100% certain but I believe that the server's event logs will capture an attempt to delete a file when the attempt fails due to permission settings. So if someone attempts to kill this log file you will know about it. I'm not sure whether that counts as a network event or a security event when the delete request comes through the network.
When you take this approach, everyone should have two entries in your shutdown logs. When you find someone who does not, take the time of that entry and talk to your security manager to find out if there is an EVENT, APPLICATION, or SECURITY entry for that time of day on the user's workstation relating to a Task Manager shutdown of an application. This is much easier if your user's aren't also the administrators of their own machine, but some sites take the easy way out on that one... If they are not the machine admins, then they probably cannot clear the event logs and that works in your favor. Anyway, that entry in the event log plus the oddball entry in your shutdown logs will be your evidence of someone doing an unkind and illegal thing to your database.
Second, as to deleting the BE, ask your security manager for help in this one. Set an AUDIT ACL on the BE file so that if anyone deletes it, you will get a silent entry in the security log. Then when you lose the file, talk to the security manager to get a printout of the log for the appropriate time. The entry will give the network name of the person performing the delete. This will be your evidence for someone deleting the BE file.
Now, the next part is trickier, but MUST be done. You say that you stand a chance to lose a day's worth of work. Whoever sponsored this database MUST own up to the need to protect it AND issue some official statement about the importance of the DB to company operations. And the cost when data gets lost because of improper use of the DB.
Then, armed with the manager's signed statement of importance and logs showing whodunit, approach the perpetrator's supervisor with your evidence. Tell the twerp in no uncertain terms that you will disable the perp's access until the twerp and the perp talk to your manager about lost time and effort due to lost transactions.
Obviously, if this database is important enough to the company, you will get results. If it is not, then you have learned something useful as well. But I can assure you that if there is value in the work done using this DB, then you will have evidence that someone cost the company the money equivalent to the salaries of those whose day's worth of transactions got lost because of a stupid user who knows just enough to be maliciously dangerous.
Now, if this persists, there is always the death-knell approach...
Keep a table in the database somewhere that tells who left the database properly. Update that via recordset operations at the start and at the end of the post-processing routines. If a person tries to come in with a record that says "You left the database improperly - cleanup is needed. Go away and try again later" then they will get the message.
In summary, the ONLY way to solve this kind of problem is to be willing to rock the boat until someone falls out. If it is YOU, then it was time to find a new job anyway. If it is the perp or the twerp, congratulations.