In this paper, we obtain estimates on the quantitative strata of the critical set of non-trivial harmonic functions u which vanish continuously on V \subset \partial \Omega , a relatively open subset of the boundary of a convex domain \Omega \subset \mathbb{R}^n . In particular, these estimates improve dimensional estimates on \{|\nabla u| =0\} both in V \subset \partial \Omega and as it approaches V \cap \overline{\Omega} . These estimates are not obtainable by naively combining interior and boundary estimates, and represent a significant improvement upon existing results for boundary analytic continuation in the convex case.