Low-energy electron microscopy (LEEM) has been used to study the structure, initial growth orientation, growth progression, and the number of layers of atomically thin hexagonal boron nitride (h-BN) films. The h-BN films are grown on heteroepitaxial Co using chemical vapor deposition (CVD) at low pressure. Our findings from LEEM studies include the growth of monolayer film having two, oppositely oriented, triangular BN domains commensurate with the Co lattice. The growth of h-BN appears to be self-limiting at a monolayer, with thicker domains only appearing in patches, presumably initiated between domain boundaries. Reflectivity measurements of the thicker h-BN films show oscillations resulting from the resonant electron transmission through quantized electronic states of the h-BN films, with the number of minima scaling up with the number of h-BN layers. First principles density functional theory (DFT) calculations show that the positions of oscillations are related to the electronic band structure of h-BN.