We investigate concentration properties of functions of random vectors with values in the discrete cube, satisfying the stochastic covering property (SCP) or the strong Rayleigh property (SRP). Our result for SCP measures include subgaussian inequalities of bounded-difference type extending classical results by Pemantle and Peres and their counterparts for matrix-valued setting strengthening recent estimates by Aoun, Banna and Youssef. Under a stronger assumption of the SRP we obtain Bernstein-type inequalities for matrix-valued functions, generalizing recent bounds for linear combinations of positive definite matrices due to Kyng and Song. We also treat in detail the special case of independent Bernoulli random variables conditioned on their sum for which we obtain strengthened estimates, deriving in particular modified log-Sobolev inequalities, Talagrand's convex distance inequality and, as corollaries, concentration results for convex functions and polynomials, as well as improved estimates for matrix-valued functions. These results generalize inequalities for the uniform measure on slices of the discrete cube, studied extensively by many authors. Our approach is based on recent results by Hermon and Salez and a general framework involving modified log-Sobolev inequalities on the discrete cube, which is of independent interest.