While all that thanatassa says is correct, and provides good tools for doing a literary analysis, I would offer another perspective, one that starts from the perspective of genre. Heinlein was a master of science fiction, and indeed, helped shape it into the field it is today. Just as detective fiction has special kinds of stories, like the locked room mystery, science fiction has common stories and story structures to which many writers have returned. In this case, the specialized sub-genre is the time travel story, specifically the time travel paradox.
In the time travel story, characters travel through time. In the time travel paradox, they travel through time and accidentally or intentionally create logical impossibilities. Heinlein’s classic story is the ultimate example of this sort of paradox. Jane is the most generic name in English (think of Jane Doe), and Jane barely has an identity. She can be read as a kind of placeholder: an anyperson, because she’s not really Jane. She’s every major character in the story, and the story can therefore be read as a kind of philosophical illustration of an idea. Jane is almost a figure in a parable.
The key to writing an effective literary analysis of "‘All You Zombies—'" by Robert A. Heinlein is understanding what is meant by "literary analysis," as opposed to other types of essay. Rather than focusing on the content of a work, a literary analysis discusses the work's uses of literary techniques, such as point of view, diction, plot structure, or figures of speech.
Thus a literary analysis of Jane's identity would focus not on her as a person but rather on the way this identity is presented to the reader. For example, one could focus on how the narration is handled. How does the story use verb tenses to deal with the various temporal shifts and paradoxes in the story? What sort of access does the narrator have to the various versions of Jane? How do pronouns reflect gender instability? How does point of view shift among the different versions of Jane?