To be more accurate, the only non showing part of a claymore's body is the center of the chest. There are theories around what's there, but no one really knows aside from Yagi.
I've always though of it as the initial point of where they start the yoma transfer into claymore candidates. Teresa said that she was "cut open" and the color drained away from her hair and eyes. It's possible that they perform a sort of pseudo-surgical procedure on them to put the yoma into their bodies. If this were true, they would have some sort of huge scar that would never have the benefits of advancing healing as it would heal naturally (see: Rafaela's left eye)
The center of the chest is the main point though. Clare told Teresa: "Doesn't this hurt?", Ruvel says to Clare: "What would that boy think if he saw your body?", and there's also the time when Ophelia scratched into Clare's chest for a "taste"