We introduce consecutive equi-$n$-squares, a variant of equi-$n$-squares in which at least one row or column forms a fixed permutation of $\{1,\dots,n\}$, taken for concreteness to be $(1,\dots,n)$. More generally, the enumeration and probabilistic arguments presented here extend to the occurrence of any prescribed permutation as a row or column of an equi-$n$-square. We derive exact and asymptotic formulas for the number of consecutive equi-$n$-squares, showing precisely how their proportion among all equi-$n$-squares rapidly approaches zero as $n\to\infty$. We also analyze the distribution of consecutive equi-$n$-squares under uniform random sampling and explore connections to algebraic structures, interpreting equi-$n$-squares and consecutive equi-$n$-squares as Cayley tables. Finally, we supplement our theoretical results with Monte Carlo simulations for small values of $n$.