  The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set
as $G$ and with two vertices adjacent if their distance in $G$ is at most 2.
Thomassen showed that every planar graph $G$ with maximum degree $\Delta(G)=3$
satisfies $\chi(G^2)\leq 7$. Kostochka and Woodall conjectured that for every
graph, the list-chromatic number of $G^2$ equals the chromatic number of $G^2$,
that is $\chi_l(G^2)=\chi(G^2)$ for all $G$. If true, this conjecture (together
with Thomassen's result) implies that every planar graph $G$ with $\Delta(G)=3$
satisfies $\chi_l(G^2)\leq 7$. We prove that every connected graph (not
necessarily planar) with $\Delta(G)=3$ other than the Petersen graph satisfies
$\chi_l(G^2)\leq 8$ (and this is best possible). In addition, we show that if
$G$ is a planar graph with $\Delta(G)=3$ and girth $g(G)\geq 7$, then
$\chi_l(G^2)\leq 7$. Dvo\v{r}\'ak, \v{S}krekovski, and Tancer showed that if
$G$ is a planar graph with $\Delta(G) = 3$ and girth $g(G) \geq 10$, then
$\chi_l(G^2)\leq 6$. We improve the girth bound to show that if $G$ is a planar
graph with $\Delta(G)=3$ and $g(G) \geq 9$, then $\chi_l(G^2) \leq 6$. All of
our proofs can be easily translated into linear-time coloring algorithms.
