No... one is definitely not being NEGLIGENT by not putting on a case or screen protector, and no, a case is not NEEDED for the phone to operate properly.
Honestly anyone who things either of the above is true does not understand logic. Negligence is defined as "failure to take proper care in doing something." If you accidentally drop your phone because you weren't paying attention, you were negligent... but not preemptively putting a case on a device is not negligence. Also... if a case were needed for the phone to work properly, than everyone here not rocking a case or screen protector would not be able to use their phone as intended... which I'm sure they are able to use their phones just fine.
I've known people with otter boxes and hard core cases and screen protectors who have shattered their screens just as many times as people who have a naked phone shatter their screens. It's honestly the luck of the drop and the angle. Sure, it might be more likely to get dings and scratches and cracks with no case or screen protector, but nothing practical will ever be able to completely secure your phone from a drop. Just be careful and not klutzy. Accidents happen, that is what insurance is for or shelling out $100 to replace the screen (or just living with it... as we can see from all the videos, the phone will still work unless you completely shred the thing...)
It all comes down to personal preference in the end... I like minimalist cases and no screen protector. I've been good for 10+ years... so I'm going to stick with it. But saying that not putting on a case is negligent or that the phone needs a case is a completely foolish argument... let people do what they want!