Fink, Mészáros and St.Dizier showed that the Schubert polynomial $\mathfrak{S}_w(x)$ is zero-one if and only if $w$ avoids twelve permutation patterns. In this paper, we prove that the Grothendieck polynomial $\mathfrak{G}_w(x)$ is zero-one, i.e., with coefficients either 0 or $\pm$1, if and only if $w$ avoids six patterns. As applications, we show that the normalized double Schubert polynomial $N(\mathfrak{S}_w(x;y))$ is Lorentzian when $\mathfrak{G}_w(x)$ is zero-one, partially confirming a conjecture of Huh, Matherne, Mészáros and St.Dizier. Moreover, we verify several conjectures on the support and coefficients of Grothendieck polynomials posed by Mészáros, Setiabrata and St.Dizier for the case of zero-one Grothendieck polynomials.