Why is important that the energy density be positive definite in field theories?
In physics experiments energy differences are the ones that are usually measurable. We cannot allow energies to be infinitely negative because then we will be measuring energy differences but with respect to what exactly?? In a system where there can be states (orbits, field configurations etc.) with infinitely negative energy we need an interpretation for all these states that we don't measure, given that usually we measure a finite spectrum of energy differences, like in the Zeeman experiment, but in such a situation we would be measuring an infinitum of energy differences that do not necessarily become smaller, because decays would be allowed to any possible state below the excited one, and there are infinite of those with unbounded distances between them.
In a system where energy densities are finitely negative, we can just change the definition of our Hamiltonian by a constant (which doesn't change the equations of motion in classical or quantum physics) so that it attains it's minimum at zero and then we have again a positive definite spectrum, but either way, the energy differences we would measure would make sense, so that constant we are adding is for convenience.
One good example for a theory with "infinitely negative spectrum" is Dirac's theory for the relativistic motion of an electron, which initially had the problem, if naively interpreted, that energies in it's spectrum are infinitely negative. Dirac justified the fact that we do not measure these states in a very solid state physics way, by asserting that they are all occupied by an electron sea, which collectively behaves as a vacuum. This assertion is just strange physically, and now we know that the Dirac equation is better interpreted as describing particles and antiparticles which have only positive energies,packaged together in one equation.