I’m sure they mean, when the chick is too undeveloped to have bones. Pippin’ (hatching) may no be exactly correct, but it points your mind in the right direction.
Pipping is the term used to describe what a chick does when it makes a hole in its shell before it hatches. At this point, the chick’s bones have not yet hardened.
Peaches when they’re blooming, they have no stone,
A chicken when its pipping, it has no bone,
The story of love, it has no end,
And a baby when its sleeping has no cryin’
So, the baby is very much alive, just sleeping and not crying. It really is a lovely little song.