We use complete enumeration of self-avoiding chains of up to N=26 monomers in two-dimensional lattices to investigate the effect of alternative implementations of backbone hydrogen bonds on the cooperativity of homopolypeptide collapse. Following a recent study on protein folding models, we use the square lattice with z=3 local conformations per monomer and lattice extensions containing diagonal steps which result in z=5 or z=7 and assume that only a subset of zh<z local conformations is compatible with hydrogen bond formation. As previously observed in heteropolymeric folding, a significant increase in cooperativity, as measured by kappa2 values, results from the coupling between hydrogen bonds and hydrophobic interactions, in such a way that hydrophobic contacts are favorable only when contacting monomers are involved in hydrogen bond formation. For some z/zh combinations the energy distribution is bimodal at the collapse transition temperature. The situation can be regarded as if all hydrophobic contacts actually decrease the energy by the same amount, 2h , with the addition of an energetic increase, epsilon2=h, as a penalty for each contacting monomer not satisfying the hydrogen bond condition. Cooperativity is little affected and might even decrease, however, when hydrogen bonds produce a decrease in energy by the same amount, epsilon1=h, for each bonding monomer. For the more general situation when the hydrogen bond effect is not equal, in modulus, to the hydrophobic interaction, i.e., epsilon2 not equalh or epsilon1 not equal h, we observe a pronounced increase in kappa2 for small epsilon2, with a maximum around epsilon2/h approximately 1.5, followed by a gradual decrease to a limiting value at large epsilon2. The opposite behavior is observed when epsilon1 is varied. The observed qualitative difference is shown to arise from opposite effects on the convexity of the total density of states of the system when subdensities corresponding to different numbers of hydrogen bonds are differently favored as opposed to the case when subdensities corresponding to different numbers of contacting monomers not forming hydrogen bonds are differently unfavored. Potential implications for the cooperativity of protein folding and protein unspecific collapse are discussed.