What if your employer says “don’t access your health records on our machine”? If you put private health information in your Twitter bio, Twitter is not obligated to suddenly treat it as if they were collecting private health information. Otherwise every single user-provided field would be maximally radioactive under GDPR.
I suspect they should say "this machine is not confidential" and have good reasons for that - you can't just impose extra restrictions on your employees just because you want to.
The law (as executed) will weigh the normal interest in employee privacy, versus your legitimate interest in doing whatever you want to do on their computers. Antivirus is probably okay, even if it involves TLS interception. Having a human watch all the traffic is probably not, even if you didn't have to intercept TLS. Unless you work for the BND (German Mossad) maybe? They'd have a good reason to watch traffic like a hawk. It's all about balancing and the law is never as clear-cut as programmers want, so we might as well get used to it being this way.
If the employer says so and I do so anyway then that’s a employment issue. I still have to follow company rules. But the point is that the company needs to delete the collected data as soon as possible. They are still not allowed to store it.
Many programmers tend to treat the legal system as if it was a computer program: if(form.is_public && form.contains(private_health_records)) move(form.owner, get_nearest_jail()); - but this is not how the legal system actually works. Not even in excessively-bureaucratic-and-wording-of-rules-based Germany.