I have one. You have one. So do bears, rabbits, squirrels, and even whales. Innie or outie, belly buttons are one of the distinguishing characteristics for mammals. They are proof that once upon a time we were tethered to an umbilical cord.
Are birds in the belly button circle? Well, get this: Inside a bird's egg there is a tiny cord that connects the developing embryo to the yolk sac. When the egg cracks open, the hatchling does in fact have a miniscule residual scar where the cord was. As the nestling grows up, the scar fades. It's pretty much invisible by the time the bird takes its first giant step from the nest.
So, yes, baby birds do have a belly button. But it's more like a temporary tattoo.