When talking about a "square deal" I think he means that they are treated fairly and equally, not treated horribly. Basically treated with some degree of respect. DuBois doesn't think that African Americans will be able to achieve a "square deal" because he says the white south wants Subordination, but the African Americans most likely will not give that to the white south. He uses the example of schools where the African American run schools may not want to answer to the "white superintendence."
That's what I concluded from the excerpt, and the best answer I could give.