I do have quite a radical view on things such as charity. I don't want to get into them too much, but you could look up Kant's distinction on perfect and imperfect duties to get an idea of how I would answer. But in the theoretical situation where a fertilized egg floated around, I would think that we ought to give it care. It would be nice if I could go Zeus style and sew it into my thigh to bring it to maturity, but as I can't, it would most probably be up to a woman. I wouldn't force her to do it, however. Again, looking at Kant's imperfect duties would explain my position.
Have you gone an volunteered to donate bone marrow? Donating bone marrow is less of an imposition than pregnancy, and you can do that as a male (unless you have some conditions yourself) Furthermore, someone with leukemia usually has a longer unassisted life expectancy than a zygote. Furthermore they usually have far more developed cognitive faculties. Perhaps organ or tissue donation should be mandatory? Looked at as a medical procedure, carrying a child to term is incredibly invasive, with months of discomfort, significant risks both short term (death very rarely) and long term (bone mass loss and diabetes.) Fetal cells persist in the woman's body for decades and have been implicated in some auto-immune disorders. In many ways, it is invasive than organ donation, which is never mandatory in the west.
Again, I agree that we do confer different rights on individuals depending on their age. To me, this is an attempt to approximate times when individuals attain certain levels of rational thinking. It's imperfect, but it's what we do. But I don't think we can extrapolate from this that there is a time when a human life has no rights whatsoever. Yes, we allow guardians to make decisions for a sick child. But I don't think we'd allow a parent to terminate the life of a perfectly healthy child. I think we would point to a child's right to life. Your example spoke of a seriously ill child. I do not think these situations are analogous.
Why are they not analogous? Consider a newborn. They cannot eat solid food or move and do not have control over their sphincters. If an adult were in this situation we would consider them very ill. In terms of being able to live an unassisted life, newborns are not healthy, they are in fact very ill, and need to be nurtured to full health over months. A zygote is even worse. It can't service any of its own biological needs unless immediately connected to the most intensive life support system know, a living woman's uterus.
A zygote is not a person. It is a human cell. It is a human cell that may be able to, under the right circumstances, develop into a new complete human animal. The only reason to think of it as something more is because it often finds itself in those very circumstances.
I challenge you to come up with a definition of "person" that satisfies the following two requirements:
1) Does not base itself on unique human dna.
2) Does not include something like a adult stem cell, which could, under the right circumstances, be induced into forming a new complete human animal.
As to why I rule out the dna aspect, I do think that thinking about this matter gets hung up on the idea of a unique DNA combination be the thing that deserves human rights. Consider three facts:
1) Twins. They clearly are separate individuals despite having identical DNA.
2) Immortal human cells. Some human cells
are viable long-term outside of a human body. These cells do not have any rights.
3) We can now successfully clone mammals. It is reasonable to think we will have the technical ability within 50 years to clone a human being. With this ability comes the fact that *any* human cell could represent the start of a new life. We can't let any human cell die then? We should clone every cell we can?
Finally, I suggest that discussing these questions in a vacuum is not appropriate, since the conclusions are not played out in a vacuum. For instance, while I completely respect your vegetarianism (I was vegetarian myself for 6 years, but not for ethical reasons.) if you think there is no animal death involved in your food, you are incorrect. If you are eating commercially produced grain for instance, you are complicit in the death of countless mice and other rodents that are killed as the fields are plowed and harvested, as well as killed to prevent them from consuming your food. I don't think you can argue that there is less suffering involved in diet of wild caught fish than in a diet of commercially produced plants.