You did not state the second half of Hillary’s statement (or I missed it). She said it was permitted to abort a fetus up to the moment before birth because the State does not grant the rights due a human being until after birth.
That is a key provision. If our human rights are a gift from the State, then her statement is true. But the USA was founded on the idea that humans enjoy unalienable human rights endowed by their Creator. Rights granted by the State can be taken away by the State.
The pro-abortionists are endorsing a viewpoint that is utterly contrary to what the USA was founded on. They have no right nor power to impose that revision on our society.