I am not sure about your first statement... In particular even with a quantum non-interacting gas with particle-in-a-box modes, the Hamiltonian is not bounded

I am not sure about your first statement... In particular even with a quantum non-interacting gas with particle-in-a-box modes, the Hamiltonian is not bounded from above and there is no reason to expect a negative temperature, no? There exist systems, like spin systems, where energy is bounded from above and so entropy decreases as you add energy, which is the definition of negative temperature... But I find it dubious that every system is such, unless I am missing something nonintuitive about say relativistic effects or so