Shaikh Ibn Uthaymeen rahimahullah said:
As for shaking hands when eating, then there is no one who wants to shake hands when eating! If a person enters and the people are eating and he wants to shake their hands, (and so) he shakes the hands of each one, (then) there is no one who does this.
[Fatawa al-Haram al-Makki 1410 H no. 8 side B]
So it is best that people do not shake hands when eating, but just exchange Salam verbally.
And Allah Knows Best
Faisal Ibn Abdul Qaadir Ibn Hassan