Does this truly mean "I eat with the hands"? Or is this simply the Irish way to say "I eat with my hands." Some languages seem to require the possessive pronouns with parts of the body, while others don't. How is it in Irish? In German you could say, "Ich esse mit den Händen" to mean "I eat with my hands." Only if you wanted to put a great deal of emphasis on them being your hands would you say "mit meinen Händen."
A Web search showed the existence of le mo lámha (e.g. Psalms 7:4 in the Irish translation), but it wasn’t used in a context of eating. I don’t think that le mo lámha vs. leis na lámha would reflect emphasis in Irish. Perhaps Irish people will respond with typical usage in IE English.
I have twice reported that "I eat with my hands" should be accepted and is more natural as an English sentence. Although there are examples of "eat with the hands" (eg you could say "Eating with the hands is not considered bad manners in this culture"), I would say that, when referring to a specific person/persons, the possessive adjective would be normal.