Inspired by the application of differential correction to initial-value problems to find periodic orbits in both the autonomous and non-autonomous dynamical systems, in this paper we apply differential correction to boundary-value problems. In the numerical demonstration, the snap-through buckling of arches and shallow spherical shells in structural mechanics are selected as examples. Due to the complicated geometrical nonlinearity in such problems, the limit points and turning points might exist. In this case, the typical Newton-Raphson method commonly used in numerical algorithms will fail to cross such points. In the current study, an arc-length continuation is introduced to enable the current algorithm to capture the complicated load-deflection paths. To show the accuracy and efficiency of differential correction, we will also apply the continuation software package COCO to get the results as a comparison to those from differential correction. The results obtained by the proposed algorithm and COCO agree well with each other, suggesting the validity and robustness of differential correction for boundary-value problems.