On Egin! is the Spanish equivalent to the French expression Bon appétit!, explained beautifully here so I need tell you no more about that.
On Egin! means literally 'that it may do you good' in Basque and is a direct translation of the Iberian Spanish expression ¡Que aproveche!. You will find the Basque version is widely used in the Basque Country instead of the Spanish term, simply because wherever possible, Basque takes prevalence.
Want to find out more? Nordicfrost has done a great job of providing 'Euskera 101' for our delight and delectation.