Can G, H, and S be negative?
G and H are like altitude.  They are not absolute; rather their values are always relative to whatever you set as a reference. So they can be either positive or negative.  There is no general constraint on the sign of their values.  For this reason, we usually talk about $\Delta G$ or $\Delta H$, since it's the changes that are meaningful; giving them arbitrary values isn't typically useful.
By contrast, there is a non-arbitrary zero for entropy.  In classical thermodynamics, it's the entropy of a pure substance at absolute zero. Since the temperature can never be lower than absolute zero in a system at thermodynamic equilibrium, $S$ (as opposed to $\Delta S$) can never be negative.
In statistical mechanics, there are various equivalent ways of defining the entropy, depending on the ensemble you are using.  The simplest definition is that for the microcanonical ensemble:
$$S = k_B \ln \Omega,$$
where $\Omega$ is the number of possible microstates of the system, and $k_B$ is the Boltzmann constant.  Since the minimum number of microstates is 1 (which happens at absolute zero), the minimum value for $S$ is zero.