The only neutral elementary particles are the neutrinos from the fermions, and the Z0 vector boson and the gluons from the bosons. None interact with the photon directly (at tree level)
All other neutral particles (e.g. neutrons, pions, atoms, etc.) have charged sub-particles in their composition, and, thus interact with photons.
We have to mention that, at elementary particle level, all the interactions are scattering. Namely, an elementary particle cannot absorb a photon, and remain the same. Composite particles that have excited bound states, may absorb a photon, and transit to an excited state. As they de-excite, they re-emit an incoherent photon.