We incorporate homogeneous traction and displacement boundary conditions into a recently proposed hybrid homogenization theory for unidirectional composites with random fiber distributions in order to investigate the convergence of homogenized moduli and local stress field statistics with representative volume element size. The hybrid approach combines elements of finite-volume and locally-exact elasticity approaches in the solution for the fiber and matrix displacement and stress fields within multi-inclusion representative volume elements that is extremely efficient and accurate, producing four-fold and greater reductions in execution times relative to the finite-volume micromechanics. Enabled by the extended approach, we then determine homogenized moduli and their standard deviations, as well as local stress field statistics, of domains containing increasing number of randomly spaced fibers under homogeneous and periodic boundary conditions for a large number of microstructural realizations. We show that fiber randomness and boundary condition type have substantially greater effects on the little investigated statistics of local stress fields than the homogenized moduli, and remain significant for domains with even 100 fibers. The choice of boundary conditions influences the location of plasticity and failure initiation at the fiber/matrix interfaces within a representative volume element, whose impact on the post-elastic homogenized response remains to be fully explored.