The first two Chandrasekhar recursions for the maximum correntropy criterion (MCC) Kalman filter (KF) have been recently derived for constant discrete-time linear systems. Their key feature is a mathematical re-formulation of the underlying MCC-based Riccati-type difference equation in terms of the involved error covariance matrix increment. Thus, the Chandrasekhar recursion-based solution is proved to yield a significant reduction of the computational complexity. This letter discusses the existence of a stable square-root solution for Chandrasekhar-type MCC-KF estimators, i.e. their computational reliability issue in a finite precision arithmetic. Two square-root solutions are proposed in terms of covariance quantities, namely within the Cholesky factorization and singular value decomposition.