The anomalous quartic neutral couplings of the.... vertex in a polarized light-by-light scattering of the Compton backscattered photons at the CLIC are examined. Both differential and total cross sections are calculated for e +e - collision energies 1500 GeV and 3000 GeV. The helicity of the initial electron beams is taken to be +/- 0.8. The unpolarized and SM cross sections for the same values of helicities are also estimated. The 95% C.L. exclusion limits on two anomalous photon couplings.1 and.2 are calculated. The best bounds on these couplings are found to be 6.85 x 10-16 GeV-4 and 1.43 x 10(-15) GeV-4, respectively. The results are compared with the exclusion bounds obtained previously for the LHC and HL-LHC. It is shown that the light-by-light scattering at the CLIC, especially the polarized, has a greater potential to search for the anomalous quartic neutral couplings of the gamma gamma gamma vertex.