The Chicago Manual of Style identifies several special situations--aside from interpolations in quotations, which we've already covered--in which square brackets should be used. In mathematical equations, square brackets are the second-level sign of aggregation:

5[(7x - 3y)(2x + 7y) + 13] = 19,880

In complex sentences involving nested parenthetical statements, square brackets identify the internal parenthetical (interestingly, this reverses the order prescribed for mathematical expressions):

"The role of chief theoretician on the Negro Question in the CPUSA fell first to Cyril Briggs (another Black intellectual from the West Indies [in his case, Nevis]) and later to Harry Haywood and James Ford."

Square brackets may enclose phonetic transcriptions of words:

"It's not difficult to see the translation of Danzig [dan(t)-sig] into Gdansk [ge-dan(t)sk]."