Chinese name of a dish containing both chicken and egg?
I think it's just invented by that one restaurant (or some restaurant else before). It's something like paronomasia, and may be seen at some Chinese restaurant, but it's not a famous/typical dish in Chinese dishes. 

親子丼 has nothing to do with the Paul Simon song. And 親子丼 is a famous/typical dish at Japan.

BTW: "Mother and Child Reunion" can be translated as "母子重逢".