  A complete list of operators contributing at the lowest order to Quartic
Neutral Gauge Boson Couplings involving photons and Z-bosons, is presented. We
show that, for the couplings we consider, the lowest order contribution is from
dimension 8 operators in the case when a light Higgs is present and from
dimension 6 operators in the higgsless case where electroweak symmetry is
non-linearly realized. We also show that these operators are generated by
exchange of the Kaluza-Klein partners of the graviton in extra-dimensional
models. We then explore the possibility of probing these couplings in the
diffractive photon fusion processes pp(\gamma\gamma \to \gamma\gamma)pp and
pp(\gamma\gamma \to ZZ)pp at the 14 TeV LHC. We find that the \gamma \gamma
\gamma \gamma-coupling can be probed most sensitively and values as small as
1/(1.8 TeV)^{4} can be measured. For the \gamma\gamma ZZ-coupling, values as
small as 1/(850 GeV)^{4} and 1/(1.9 TeV)^2 can be probed in the light Higgs and
higgsless cases respectively, which is an improvement by orders of magnitude
over existing limits.
