From page 33 of the PHB it says that they "have personal names given at birth." In my mind birth would mean live young. Had it said hatching then I would think of eggs.
In the next sentence it says that "a childhood name* ... is often used among clutchmates." I have only ever heard clutch used for creatures born from eggs. Live young tend to be called a litter.
Unless I am mistaking the words used, a case could be made for both situations. I personally prefer to think of them hatching from eggs instead of being born live, but I would be willing to accept either if the DM had a preference.
*As opposed to the name they receive at adulthood.