Piece or slice is fine, you just want some meat. A slice means it's neatly and finely cut. A piece could also mean this, but you could also receive a chunk of meat.

both sentences are okay yet, "May I have a slice of your meat?" is more conventional. :)

"May I have a piece of your meat" or "May I have a slice of your meat" Both are fine but just take note of the spelling :D "piece" is different from "peace".