Two of the most eloquent supporters of rights based on egoism were Max Stirner and Ayn Rand. To Rand, the fundamental ethical distinction is whether the beneficiary of conduct is the actor, or someone or something other than the actor - egoism or altruism.
A "right" is a moral principle defining and sanctioning a man's freedom of action in a social context. There is only one fundamental right (all the others are its consequences or corollaries): a man's right to his own life. Life is a process of self-sustaining and self-generated action; the right to life means the right to engage in self-sustaining and self-generated action - which means: the freedom to take all the actions required by the nature of a rational being for the support, the furtherance, the fulfillment and the enjoyment of his own life. (Such is the meaning of the right to life, liberty and the pursuit of happiness.)
Note that Rand assumes a universality principle - that all men have the same moral perogatives. This is basically the same as the the categorical imperative or the Golden Rule.
The concept of a "right" pertains only to action - specifically, to freedom of action. It means freedom from physical compulsion, coercion or interference by other men.
Thus, for every individual, a right is the moral sanction of a positive - of his freedom to act on his own judgment, for his own goals, by his own voluntary, uncoerced choice. As to his neighbors, his rights impose no obligations on them except of a negative kind: to abstain from violating his rights. - Ayn Rand, "Man's Rights"
Max Stirner explicity rejected such a principle. But in one interpretation of his main essay, "The Individual and His Property" (also translated as "The Ego and His Own"), he claims that, as least among the enlightened rational egoists, there will be a "union of egoists" which will, in effect, treat each other as if they had rights, in the mutual understanding that everyone uses everyone.