Can a zeroth order reaction be reversible?
A zeroth order reaction does not achieve equilibrium as a zeroth order reaction.  Instead the kinetics will go away from zeroth order when we get close to the equilibrium condition.

Zeroth order kinetics is an approximation we get to when some aspect of the reaction approaches a saturation condition, for example the reaction mechanism may involve adsorption on a surface and there is enough reactant to saturate all the adsorption sites.  Such conditions are perforce relatively far from equilibrium; "saturation" and "reversibility" do not come about simultaneously.