What exactly is the requirement (in terms of saturation) for a reaction to be at equilibrium?
If you add solid sodium chloride to a saturated sodium chloride solution, it is still at equilibrium. If you increase the sodium ion concentration or the chloride ion concentration (by e.g. adding a high concentration of NaOH or LiCl), then the solution is supersaturated and will not be at equilibrium.


  What exactly is the requirement (in terms of saturation) for a reaction to be at equilibrium?


A reaction is at equilibrium when the reaction quotient is equal to the equilibrium constant. For a precipitation reaction, the solid species is not part of the equilibrium constant expression, so adding more solid does not change the situation. Changing the concentration of the ions in solution, though, will change the reaction quotient, and the reaction will no longer be at equilibrium.