Then if it shows a pheasant beak, you'd know it's a pheasant hybrid, ruling out a male Coturnix. I just don't see it though—looks like a normal chicken chick, maybe a little skinny.
Pheasant hybrids seem to keep the pheasant's beak shape. And "To our knowledge, hybridization by reverse crosses (female chickens × male quails) has never been successful." Via...
What I'm saying is it had to be a rooster. Maybe it's not even from this hen's egg, if you're sure she couldn't have encountered someone else's male. But it's certainly a chicken chick.