I do not see these two as being of the same genre, but their similarities, in general, may be what your son's teacher is looking for. I hope the following helps.
The definition of "genre" according to dictionary.com is:
a class or category of artistic endeavor having a particular form, content, technique, or the like: the genre of epic poetry
When we think of genres, we think of books that are similar in nature or topic: mystery, romance, children's fiction, etc.
Perhaps the use of the word "genre" is used here in a more general way: classifying these two pieces of literature in what they have in common (I am unaware of the name of a genre that would reflect this).
However, in reviewing Dr. King's "Letter From Birmingham Jail," and Sophocles' Antigone, I find that the commonality is not in the kind of literature it is, but in the common themes both pieces present to the reader.
The plot of the play Antigone reflects a woman (a minority at the time) controlled by social, political, religious, and gender-related norms of the era who is struggling to do what is ethically right. She is demanding that King Creon allow her to bury her brother Polyneices (who has been judged a traitor); if she cannot do so, the Greeks believed his soul would wander the Earth for a hundred years before finding its rest. As a woman, she has no power or influence. If searching for discrimination, I would hazard to guess that had she been a man, she might have had more success with the leaders of the male-dominated society of which she was a part.
In "Letter From Birmingham Jail," Martin Luther King, Jr., also a member of a minority, implores the political and social leaders of the 1960s, to do the ethical and moral thing by allowing blacks equal rights under the law. His concerns would not only have lifted up the unfair treatment of blacks in terms of prejudice and segregation, but more seriously with the murder of civil rights activists and innocents killed due to the color of their skin, and nothing more.
The similarities of the content of the two pieces are what make me see them as products of a genre that challenges the status quo and demands that appropriate actions take place regarding what is decent, ethical and morally correct.
Antigone never gets the answer she wants. Dr. King, however, continued to fight for equality, and his "dream" was realized with the signing of the Civil Rights Act in 1964, which outlawed discrimination and segregation against blacks—and women.
For three similarities: each piece centers around the endeavors of a member of a minority group; both central figures are trying to find justice; both figures are at a disadvantage because they lack power and credibility among those in control at the time they lived because they were minorities.
I hope this is of some assistance. Best of luck.