This isn't really an answer, so I'll post it as a comment instead.
Constrained linear regression is a more difficult problem than ordinary least squares regression. (It would be nice if the Optimization Toolbox had some statistical functions, especially since statistics for constrained optimization are different than for the unconstrained optimization used in the Statistics Toolbox.) Constrained estimators and tests in the multiple linear regression model (Part I) discusses the theory and calculations for the constrained linear regression covariance matrix. Note that the X matrix is likely the design matrix (that incliudes a column of ones) although it doesn't specifically mention that in this particular lecture.
