We consider random right-angled Coxeter groups, $W_Γ$, whose presentation graph $Γ$ is taken to be an Erdős--Rényi random graph, i.e., $Γ\sim \mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of these random groups. We resolve a conjecture of Susse and determine the connectivity threshold for square percolation on the random graph $Γ\sim \mathcal{G}_{n,p}$. We use this result to determine a large range of $p$ for which the random right-angled Coxeter group $W_Γ$ has a unique cubical coarse median structure. Until recent work of Fioravanti, Levcovitz and Sageev, there were no non-hyperbolic examples of groups with cubical coarse rigidity; our present results show the property is in fact typically satisfied by a random RACG for a wide range of the parameter $p$, including $p=1/2$.