  A regular nilpotent Hessenberg Schubert cell is the intersection of a regular
nilpotent Hessenberg variety with a Schubert cell. In this paper, we describe a
set of minimal generators of the defining ideal of a regular nilpotent
Hessenberg Schubert cell in the type $A$ setting. We show that these minimal
generators are a Gr\"obner basis for an appropriate lexicographic monomial
order. As a consequence, we obtain a new computational-algebraic proof, in type
$A$, of Tymoczko's result that regular nilpotent Hessenberg varieties are paved
by affine spaces. In addition, we prove that these defining ideals are complete
intersections, are geometrically vertex decomposable, and compute their Hilbert
series. We also produce a Frobenius splitting of each Schubert cell that
compatibly splits all of the regular nilpotent Hessenberg Schubert cells
contained in it. This work builds on, and extends, work of the second and third
author on defining ideals of intersections of regular nilpotent Hessenberg
varieties with the (open) Schubert cell associated to the Bruhat-longest
permutation.
