Physical meaning of the Ising Hamiltonian

Negative energy is extremely common in nonrelativistic physics, there is nothing conceptually different about it than positive energy. I think it would be helpful to see a particularly simple example: in classical mechanics, the potential energy between two gravitating bodies is $$U(r) = -G frac{m_1 m_2}{r}$$ The physical content of this equation should be clear: the attraction between the two bodies is proportional to each of their masses, $G$ is a proportionality constant which fixes the units, and the energy decreases as the two bodies become closer (ie, as $r$ decreases, $U(r)$ becomes more negative). Here, it makes perfect sense that the energy is negative: it is sensible for $U(r)$ to be zero when the two bodies are infinitely far apart, so it must be negative for the energy to decrease as the bodies attract each other.

As for the Ising model, you are probably looking at Hamiltonians like $$H = -J sum_{langle i, j rangle} sigma_i sigma_j $$ Indeed, you can see by inspection that the energy is negative when all the spins point in the same direction (ie, $sigma_i = +1$ for all $i$ or $sigma_i = -1$ for all $i$). If t his really bothers you, and you cannot accept negative energies, then you can equally well choose the Hamiltonian $$H' = -J sum_{langle i, j rangle} (sigma_i sigma_j - 1)$$ which you can see by inspection differs from $H$ only by a constant. In statistical physics, an overall constant of energy does not affect any observable quantities, so you are free to choose either. In other words, there's no experiment that will tell you the absolute energy in the system, only differences, and this constant will therefore always cancel.

Perhaps it's worth remarking on why you might feel negative energy is so problematic -- I think this is possibly due to popular physics creating misconceptions. Indeed, there are situations where a "negative energy" would be really problematic in physics. The best example is general relativity, where absolute energies are indeed meaningful. But this is very far-afield from the Ising model, where you simply do not need to worry about negative energies.