The self-couplings of the electroweak gauge bosons are completely specified by the non-Abelian gauge nature of the Standard Model (SM). The direct study of these couplings provides a significant opportunity to test the validity of the SM and the existence of new physics beyond the SM up to the high energy scale. For this reason, we investigate the potential of the processes gamma gamma -> ZZ, e(-)gamma -> e(-)gamma*gamma -> e(-)ZZ, and e(+)e(-) -> e(+)gamma*gamma*e(-) -> e(+)ZZe(-) to examine the anomalous quartic couplings of ZZ gamma gamma vertex at the Compact Linear Collider (CLIC) with center-of-mass energy 3 TeV. We calculate 95% confidence level sensitivities on the dimension-8 parameters with various values of the integrated luminosity. We show that the best bounds on the anomalous f(M2)/Lambda(4), f(M3)/Lambda(4), f(T0)/Lambda(4), and f(T9)/Lambda(4) couplings arising from gamma gamma -> ZZ process among those three processes at center-of-mass energy of 3 TeV and integrated luminosity of L-int = 2000 fb(-1) are found to be [-3.30; 3.30] x 10(-3) TeV-4, [-1.20; 1.20] x 10(-2) TeV-4, [-3.40; 3.40] x 10(-3) TeV-4, and [-1.80; 1.80] x 10(-3) TeV-4, respectively.