The set of all $m\times n$ Boolean matrices is denoted by ${\mathbb{M}}_{m,n}$. We call a matrix $A\in {\mathbb{M}}_{m,n}$ regular if there is a matrix $G\in {\mathbb{M}}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${\mathbb{M}}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $min\{m,n\}\le 2$, then all operators on ${\mathbb{M}}_{m,n}$ strongly preserve regular matrices, and if $min\{m,n\}\ge 3$, then an operator $T$ on ${\mathbb{M}}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T\left(X\right)=UXV$ for all $X\in {\mathbb{M}}_{m,n}$, or $m=n$ and $T\left(X\right)=U{X}^{T}V$ for all $X\in {\mathbb{M}}_{n}$.