No, a photon cannot have zero spin. A photon is a particle of light, and it always has a spin of 1. However, its spin can only take certain values depending on the direction it's moving. Photons are
massless particles, so they don't have the full range of spin values that particles with mass might have.
The spin of a photon is always
1, but because it's massless, it can't be in the spin states that would correspond to higher spin values like a particle with mass could. In fact, for a photon, the spin state is always described by two possible orientations (often called left-handed and right-handed circular polarization, or simply two helicity states): +1 and -1. These correspond to the photon spinning in one direction or the opposite direction relative to its motion.
So, in summary, a photon always has spin 1, but it can't have a spin of 0.