Intuitionist Michael Huemer has written a long article on why he is not an Objectivist, despite his basic agreement with most of its conclusions. While I disagree with Huemer that Objectivism is basically correct, I think his criticism is logical and well argued.
If egoism is true, then the sole justification for my doing or refraining from doing anything is that it serves my interests. By the same token, it must be said that the sole justification for any other person’s doing A or refraining from A is that it serves his interests.
Now how do rights fit in here? To say that I have a right to A, where A is something I can either do or have (as in “I have a right to free speech” or “I have a right to own a gun”) is to say that it is morally wrong for others to forcibly interfere with my doing or having A. It is to say something about what is morally right for other people to do with respect to me. (It doesn’t constrain my actions; if I have a right to do A, I may still interfere with my own doing of A.)
Now how can I defend my rights intellectually? How do I show that I have a right to do something? If egoism is true, in order to show this, I would have to show that it is in the interests of other people to allow me to do A! This seems outrageous from an individualist ethics point of view, but the consequence strictly follows: if egoism is true, the only possible justification for claiming that other people should do X would be that it serves their respective interests to do X; so the only justification for claiming that other people should not interfere with my doing A is that it is in others’ interests not to interfere with my doing A.