What is the ruling on shaking hands while eating? And is there any evidence for this in Islamic law?

If the intention is to shake hands during a meal, then the jurists have not mentioned anything about it due to the hands being occupied. However, if the intention is to give salutations (saying "peace be upon you"), some jurists have deemed it disliked, especially if the morsel is in the eater's mouth, and in this case, the one who gives salutations does not deserve a reply. But if he is eating and the morsel is not in his mouth, then there is no harm in giving salutations, and returning the greeting is obligatory.