Yes, I feel it is every employee's responsibility to work safe. My years working for UPS has demonstrated one thing: the company will talk all about safety, but they will do very little to make this happen, unless you hold their nose directly to it. When management has the choice between safety and production, production always gets the nod. Now, they won't admit it, but any driver on this forum will testify that when the streets are icy, or buried in snow, and their truck is loaded, and the decision is between getting done and bringing that load back, management will persuade you to get those packages delivered. It was always funny how they tip-toed around the safety issue. Clearly, we can see and feel what the priority is.
That is why, yes, safety is on the employee. No, I don't think an employee should be fired if something happens where an employee gets hurt. If an unloader gets hurt because a wall of packages falls on him/her, should he they be fired, as you claim? Of course not. Using your logic, if an employee gets hurt, then the immediate supervisor should be fired too, right? After all, the immediate supervisor has responsibility over his workers, just like he does with their production, so it is only logical for that supervisor to be terminated too, right?
No, what I'm talking about is taking care of yourself. Meaning, if you have to make a choice between making service and working safe, you go safe, every single time. Even if that means missing packages or not getting a load delivered on time; if that means you work safe at all costs, you work safe.
In feeders, I drive on a, sometimes, snowy freeway. If I feel it is unsafe to continue to drive, I will shut everything down and park it. If that means both of my trailers misses the sort, then that is what working safely means to me. Now when I call dispatch and tell them I'm shutting it down, they get angry. And this is what I'm talking about when I say safety is something the sups talk big about, but in practice, it is a paper tiger.
And if you get injured, you will lose money, maybe even hurt your body permanently. If you're off work, UPS will simply fill your spot. They won't miss you or the job you do for them. You have to look out for yourself. No matter what the cost.
So, to answer your question, I, I believe every employee should take care of his/her own safety.