"accidentally turning it on"? Really? There's, like 9 key strokes involved in that operation - how in the world are they doing this accidentally?
No, you can't disable the alternate greeting such that it can't be turned on by a subscriber that's signed into the mailbox.
You can use a call handler for this and have the messages delivered to an individual subscriber or a DL such that folks aren't signing into a subscriber used for the greetings - that would solve the issue at hand anyway.