We provide upper bounds on the chromatic number of the square of graphs, which have vertex ordering characterizations. We prove that G(2) is (3 Delta - 2)-colorable when G is a cocomparability graph, (Delta + mu)-colorable when G is a strongly orderable graph and (Delta + 1)-colorable when G is a dually chordal graph, where Delta(G) is the maximum degree and mu(G) = max{vertical bar N-G(x) boolean AND N-G(y)vertical bar: x, y is an element of V (G)} is the multiplicity of the graph G. This improves the currently known upper bounds on the chromatic number of squares of graphs from these classes.