  Dunkl operators may be regarded as differential-difference operators
parameterized by finite reflection groups and multiplicity functions. In this
paper, the Littlewood--Paley square function for Dunkl heat flows in
$\mathbb{R}^d$ is introduced by employing the full "gradient" induced by the
corresponding carr\'{e} du champ operator and then the $L^p$ boundedness is
studied for all $p\in(1,\infty)$. For $p\in(1,2]$, we successfully adapt
Stein's heat flows approach to overcome the difficulty caused by the difference
part of the Dunkl operator and establish the $L^p$ boundedness, while for
$p\in[2,\infty)$, we restrict to a particular case when the corresponding Weyl
group is isomorphic to $\mathbb{Z}_2^d$ and apply a probabilistic method to
prove the $L^p$ boundedness. In the latter case, the curvature-dimension
inequality for Dunkl operators in the sense of Bakry--Emery, which may be of
independent interest, plays a crucial role. The results are dimension-free.
