I would first argue that the use of the word "employment" may not be best. Employment entails that a job is assigned to you and that you receive an income for completing said job. This is not necessary in order to do "work". Take household chores, for example, which are certainly work but aren't assigned to you and don't give you income.
I think more generally, the phrasing which makes the most sense is "must people do work in order to have self worth and dignity?" This may not look very different, but it allows us to talk about the meaning of work in general, as well as different types of work.
What is work? There are certainly different ways to answer this question, particularly in terms of suffering or labor. But I think the most general answer is: work is what must be done in order to complete a task. That is, there is a task, say doing the dishes, and there is work-the actual process of doing the dishes. It's analogous to talking about a journey (work) and its destination (completed task).
From a perspective of evolutionary biology, it makes sense that we would have a positive response to at least some amount of work. It takes work to survive; in order to get food from hunting, the process of hunting, itself, must be done.
More generally however, I would argue that doing work is deeply related to the human search for meaning. As you hinted at in your question, it's true that humans would have many emotional and social problems if we only had completed tasks but no work. The philosopher Philipp Mainlander spoke a lot on this. He essentially said that there exists "the will" in all living things which drive it to "suffer". Mainlander defined suffering as the work necessary to maintain structure. That is, if we were to do no work (no suffering), then we would die. For example, we must eat and sleep to live. These are both things we have to do to survive, yet they take time and energy and we may not want to do them. In truth, there is no fundamental reason why we ought to succumb to the will and continue suffering; it is simply an aspect of life and any objective reason for it can't be determined. This then plays into existentialist philosophy, in that we must find our own reasons-if an objective reason is indeterminable, then a subjective one should be found instead.
Mainlander's beliefs would eventually be supported by modern science. It is simultaneously both true that living things must exert energy (do work) in order to maintain their structure and that they would like to conserve energy as much as possible. This contradictory nature of life leads to weird phenomena that we are all familiar with. For example, we have evolved to require a certain amount of physical exertion to be "well". This leads to the absurdity of the modern day where we simulate the fatigue of surviving in the wilderness (working out) in order to serve our outdated biology. We don't need to do this work to survive, but we've evolved to only be mentally well if we do.
To summarize: Living things must do work to survive such that, even in a world where that isn't immediately true, we've evolved as if it is. So, we need to do work in order to be well, regardless of the modern conditions for survival. Furthermore, there is no reason why we ought to continue doing work just so that we can survive to do more work. Therefore, in the face of emptiness and death, we must find our own personal reasons to do so-our own meaning.
To connect this more directly to your question, dignity and self worth are traits that are achieved by the fulfillment of meaning. In order to be well, we each have personal values. These values must be understood through intense, deep self discovery and are entirely your own. The satisfaction of these values can be equated to the fulfillment of meaning, in a semantic sense. The satisfaction of these values however, is a task, and tasks require work.
In conclusion, the answer is yes, but only conditionally. In a kind of circular sense, we must do work to satisfy our values in order to be well, and our values relate to our meaning which justifies why we ought to do work to be well in the first place. So yes, we must do work in order to have dignity and self worth, but only work which gives us meaning. That is, we should only do work which contributes towards the satisfaction of our values, but should not do work that doesn't. Doing work which does not give you meaning will not give you dignity and self worth, and ought to be avoided at all costs, for if your life is spent doing things which do not justify living, then your life is worthless in your own eyes-from the perspective of your own meaning.