I feel very strongly about this issue, and it way exceeds Church teachings. I don’t think atheists should be donating eggs (or sperm!) in any anonymous situation, because those donations enable single people to become single “parents,” which is already an epidemic in our country, and world. If these egg and sperm banks had been regulated long ago, or were there an avenue to make them subject to legislation (accessible only to heterosexual married couples), then I might feel differently about non-Catholic donations. But since all they are now, i.m.o., is Brave New World laboratories disconnected from donors, and since these donations are abused so often to create fatherless or motherless children (like we don’t have enough of those now

), I would be appealing to any faith-less daughter, or son, to consider societal consequences.
I have single friends and acquaintances who recently sought this route because they wanted to “experience parenting”

, and frankly even they admit that the results have been extremely disappointing. They thought that single parenting would be very different than what it turned out to be, and all these people are even financially well off. (Interestingly, I also have single friends who adopted children – some from overseas, some domestically – and they are happier about their situations, possibly because they did not enter parenthood from the standard of a romantic ideal of a physically ideal baby.)
Not having a mate makes a huge difference for a single parent, and a gigantic difference for a child. So again, I would be appealing to any faith-less child to consider the children that result from these anonymous donations.
That’s on top of all the Catholic teaching about it, which is, in a nutshell: we do not separate our reproductive activity from our personhood. To do so is a grave offense against life. Life, and reproducing that life, is to be seen in the context of a particular couple’s love for each other (at best), or (secondarily) the growth within the woman herself, should she find herself pregnant in less than ideal circumstances. Anything that reduces, yet worse – eliminates – the proprietary aspect of that reproduction, is something that is forbidden to Catholics and should be seriously regarded by non-Catholics as well.