The following quotations from Rand can be found here, together with references.
An embryo has no rights. Rights do not pertain to a potential, only to an actual being. A child cannot acquire any rights until it is born. The living take precedence over the not-yet-living (or the unborn).
Abortion is a moral right—which should be left to the sole discretion of the woman involved; morally, nothing other than her wish in the matter is to be considered. Who can conceivably have the right to dictate to her what disposition she is to make of the functions of her own body?
If Ayn Rand weren't so popular among adolescents of all ages, if she were an unknown as opposed to a well-known hack, I wouldn't be wasting time refuting this nonsense. But she is very influential, so it is worthwhile exposing her incoherence. If you complain that my tone is harsh and disrespectful, my reply will be that it is no more harsh and disrespectful than hers is: read the quotations on the page to which I have linked. He who is strident and polemical will receive stridency and polemic in return. You reap what you sow.
In the first paragraph above Rand equates the unborn with the not-yet living. This implies that a third trimester fetus is not living. What is it then? Dead? Or is it perhaps neither living nor dead like an inanimate artifact? Obviously, a human fetus is a living biologically human individual. Obviously, one cannot arbitrarily exclude the pre-natal from the class of the living -- unless one is a hack or an ideologue.
Let me expand on this just a bit. One cannot answer philosophical questions by terminological fiat, by arbitrarily rigging your terminology in such a way that the answer you want falls out of the rigging. Would that Rand and her followers understood this. My post Peikoff on the Supernatural carefully exposes another egregious example of the shabby trick of answering philosophical questions by terminological fiat.
Now consider the enthymematic argument of the first two sentences of the first paragraph above. Made explicit, it goes like this. (1) Rights do not pertain to a potential, only to an actual being. (2) An embryo is a potential being. Therefore, (3) An embryo has no rights.
A being is anything that is or exists. So if x is a merely potential being, then of course it cannot have any rights. A merely potential being is either nothing or next-to-nothing. But a human embryo is not a merely potential being; it is an actual human (not canine, not lupine, not bovine, . . .) embryo. Indeed it is an actual biologically human member of the species homo sapiens. That is a plain fact of biology. So the second premise is spectacularly false.
If Rand were to say something intelligent, she would have to argue like this:
(1*) Rights do not pertain to potential persons, only to actual persons. (2*) An embryo is a potential person. Therefore, (3) An embyo has no rights. Unlike Rand's argument, this argument is worth discussing. But it is not the argument Rand gives. I have countered it elsewhere. See Abortion category.
The second paragraph quoted above is as sophomoric as the first -- if that's not an insult to sophomores. It is a clumsy gesture in the direction of what is often called the Woman's Body Argument. Follow the link for the refutation.