We prove a criterion of when the dual character $χ_{D}(x)$ of the flagged Weyl module associated to a diagram $D$ in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $χ_{D}(x)$ are either 0 or 1. This settles a conjecture proposed by M{é}sz{á}ros--St. Dizier--Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink--M{é}sz{á}ros--St. Dizier and Hodges--Yong, respectively.