Shaikh Ibn ul-‘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 nd he wants to shake their hands. He shakes the hands of each one, 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 Qaadir Ibn Hassan