From The New York Times: “At this point, everybody is on their toes,” said Stanley L. Johnson, a spokesman for the North American Electric Reliability Council, an industry group in Princeton, N.J.
Harbrace Chapter 6: Agreement says (in part): Make a pronoun agree in number with its antecedent [the word coming before the pronoun that the pronoun signifies].
Antecedents like "everyone, " "someone, " "everybody," "each," "either" are singular.
In the sentence above, Mr. Johnson could have replaced "their" with "his or her," or he could have said, "At this point, all control room engineers are on their toes."
Note: "None" used to be treated strictly as singular, because it is a contraction of "not one," but these days, says Harbrace 6a(7), "When used as subjects, "all," "any," "some," and "none" may take either a singular or a plural verb, generally depending on the context.