I've done a little research on this and studies have shown that flash does not hurt the eyes of baby animals or even baby humans. Most animals that are hatched/born with their eyes open have a good blinking reflex. I take photos of baby birds often and I worried about it, too.
Just to be on the safe side, though, I try to limit my flash to as little as possible. Also, I don't deliberately use flash on chicks/ducklings that are hatching or still wet from hatching. I want to make sure they have had some exposure to natural light, first. I did, however, observe some baby killdeer (a little brown plover) who hatched right in the middle of the fireworks show at the lake. They showed no sign of vision problems later on and can see, probably 1000 times better than I do.