  We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every
such graph either has a clique cutset, or a universal vertex, or belongs to
several special classes of graphs. Using this result, we show that for any
$(P_6,C_4)$-free graph $G$, $\lceil\frac{5\omega(G)}{4}\rceil$ and
$\lceil\frac{\Delta(G) + \omega(G) +1}{2}\rceil$ are tight upper bounds for the
chromatic number of $G$. Moreover, our structural results imply that every
($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and
thus the existence of a polynomial-time algorithm that computes the chromatic
number (or stability number) of any $(P_6,C_4)$-free graph.
