I think we need to ask the important question... Since Tengu women have anatomy that appears identical to Human women, do they have a uterus/womb?
If so, then Tengu do not lay eggs, but rather carry their young inside their womb until birth. Based on what should be fairly common sense, and despite having some bird-like traits, Aya appears to be a mammal in every sense. She has wider hips to allow childbirth, breasts to allow breastfeeding of young, and appears to, at least on the outside, have a Human-like sexual anatomy. Plus, the size the egg laid would need to be would be far too large to be practical. The egg would need to be large enough to allow a baby sized Tengu child to form inside of it, else it would likely be too weak to nurse and survive outside the egg once hatched. The energy required to form an egg that large, then push it out of her body, make it seem unlikely.
Without canonical intervention from ZUN, I think it is probably safe to assume that Tengu do not lay eggs like a bird, but rather carry their offspring inside of their womb, then give birth vaginally, like Humans do.