Stationary expansion shocks have been identified recently as a new type of solution to hyperbolic conservation laws regularized by nonlocal dispersive terms that naturally arise in shallow‐water theory. These expansion shocks were studied previously for the Benjamin‐Bona‐Mahony (BBM) equation using matched asymptotic expansions. In this paper, we extend the BBM analysis to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow‐water equations. The extension for a system is nontrivial, requiring a combination of small amplitude, long‐wave expansions with high order matched asymptotics. The constructed asymptotic solution is shown to be in excellent agreement with accurate numerical simulations of the Boussinesq system for a range of appropriately smoothed Riemann data.