Is it possible to have a photon with -1 spin?

By definition of spin $S$ it is a positive integer number or zero. Not to confuse with the spin projection possible values $S_z$, which may run from $-S$ to $S$.

The mention of the spin of elementary particle is closely related to the representations of the Poincare group, since the mathematical definition of the elementary particle is irreducible representation of the Poincare group. Our world within some approximation obeys the special relativity, and the above definition of the elementary particle just reflects this fact.

Massless particles, as the representations of the Poincare group, are characterized not by spin, but by helicity $\lambda$ - the projection of total angular momentum on the direction of motion. Formally this is because the Casimir operator, which defines the spin of representation (squared Pauli-Lubanski operator), is always zero for massless representations, and, also, the little group for massless Lorentz orbit is ISO(2) group. Physically this can be understood from the meaning of the spin - it is the total angular momentum at rest, and therefore can't be defined for massless particles moving with the speed of light.

Now let's discuss your question. In general, irreducible states with helicities $\lambda , -\lambda$ belong to different representations, and hence to different elementary particles. This is because there is no continuous transformation (of the Poincare group) which can convert $\lambda$ state to $-\lambda$ state. This is the huge difference in compare with case of non-zero mass, for which the representation with given spin $s$ includes helicities $-s,-s+1,...,s$ states initially, since there exist continuous transformation, which relates these values of helicities.

Note, however, that state with helicity $\lambda$ can be converted into the state with helicity $-\lambda$ by discrete transformations of the Lorentz group, for example, by spatial inversion transformation. So if the given theory is parity invariant, we have to introduce the particle, which corresponds to the direct sum of representations with $h = \pm\lambda$. The EM theory is an example of parity invariant theory, and corresponding interaction mediator - photon - thus has two possible helicities.

Spin 1 just means that the spin in any direction can assume values out of {-1,0,1}. The 0 is only possible for massive particles, so the photon can have spin -1 or +1. That's like clockwise and anticlockwise circular polarization