haha, this is funny. Actually you can use both forms to say "eat breakfast". Desayunarme is correct, (although we not use "comerme" or "cenarme" with the same meaning), but it's used in coloquial speech.
As you said, desayunarme could also mean "eat myself for breakfast", but I think you wouldn't hear that meaning too often, unless you read a blog of a person living in a desert island!! lol