Question: Relevant Objectivist terms and definitions, as I understand them, are:
EXISTENT: something that exists (and has a physical basis)
UNIT: an existent regarded as a separate member of a group of two or more similar members
CONCEPT: a mental integration of two or more units which are isolated according to a specific characteristic(s) and united by a specific definition
Q: How do Objectivists treat notions of non-existents?
Q: What terms do they use to reference them—for apparently "concept" is not allowed?
Q: Why would they not allow the term "concept" to refer to existents and non-existents, for the sake of discussion, and, only when important, accompany such discussions with declarations of existence and non-existence.
Apparently notions of non-existents cannot be called "concepts", which seem to be reserved for things with physical qualities. And yet, the same intellectual mental abstraction abilities are needed to understand and communicate them.
For example, take an intellectual story-teller who teaches "deep" lessons to a large audience in a fictional format. The story-teller may mentally invent notions of a fictitious race of beings, with plenty of complex made-up qualities, who are distinguished from other fictitious beings.
The audience debates many aspects of the story, about what can be inferred, when, etc. To make progress in discussions, they need to form "concepts" about the fictional races, in the same way that they would if they were discussing existing races of beings.
But is it true that Objectivists wouldn't allow the audience to call such notions "concepts"? What should they call them?
Answer: "Existent" as Objectivists use the term has no implication of "physical" or "material." Mental contents are existents. Fictional inventions are existents in this sense, although they don't exist outside the mind. Objectivists normally distinguish the physical from the mental, but both cases categorize existents.
Objectivists normally call concepts of fictitious things, such as "unicorn," concepts. Just that.
Similarly, the concept of a "character" (i.e., of a fiction work) is just a concept. The existents such concepts integrate are peculiar, in that they are essentially imaginary and exist as such only in the minds of those who imagine them.
I don't know of a well-developed Objectivist theory on how such concepts relate to concepts of physical entities, actions, relations, and attributes. Going out on a limb here, I would imagine that most concepts of the imaginary function in effect like abstractions from abstractions: we only identify a unicorn, for example, by its horn. We aren't, indeed, really sure what other traits one might have—or might not have. Is "Acorna, the unicorn girl," in the Anne McCaffrey novel of the same name really a unicorn? Hmm... who could say? On what grounds? The only way to settle the matter would be in reference some tradition of unicorn stories in which other properties (like similarity to horses) are imputed to unicorns. The units of these concepts are themselves abstract and imaginary.
This is an interesting subject for further study, but not a real problem for the basic Objectivist approach to concepts.