When American robin chicks fledge, meaning leave the nest, it is pretty much empty. It's also fairly clean, as the parents have been removing the waste from the babies after each feeding.
Aside from the infrequent finding of a piece of blue shell, the eggs are notably absent once the birds have fledged.
What happens to the eggs?
It's true, egg shells are edible, and birds will eat them. Mommy robin probably snacked on the shells while she was sitting on her babies keeping them warm. After laying their eggs, birds are rather depleted of calcium, and eagerly eat the shells up to replenish those minerals.
Eggs shells are actually a recommended food for bird feeders. Bird watchers will lightly bake chicken egg shells to make them brittle, then put them in their bird feeder.
