"Any club cannot use . . ." vs. "No club can use . . ."

You are correct. Any ... cannot sounds very wrong in my ears

You can however use this construct:

Please note that the copy machine cannot be used by any club/group after 8:00pm.

or

Please note that after 8:00pm, the copy machine cannot be used by any club/group

I apologise that I cannot find any rules you can use to explain this to them.

Perhaps someone else can

The nearest I can get to any kind of rule derives from the distinction made in the ‘Longman Student Grammar of Spoken and Written English’ between ‘assertive’ forms such as some and non-assertive forms such as any. The authors write that ‘non-assertive forms follow the not'. This would allow the passive construction:

The copy machine cannot be used by any group after 8:00pm.

but would not allow the active construction in your example, where any precedes (can)not.

This explanation might need testing. A single counter-example would undermine it.