Existentialism is not a very unified school of thought. The main existentialists—Kierkegaard, Nietzsche, Heidegger, and Sartre—disagree among themselves on many fundamental philosophical questions. Consequently, it is very difficult to answer your question. However, it is accurate to say that there are meaningful similarities between Ayn Rand
's thought and the thought of both Nietzsche and Sartre. In both these case, though, there are also some very significant differences. Rand's views are not especially similar to Kierkegaard's, and they are completely and fundamentally opposed to Heidegger's.
Let us consider some examples:
Nietzsche - Rand
Nietzsche and Ayn Rand
are both highly critical of the moral ideals of Christianity. They both think that Christian ideals have had a negative influence on secular ethical thought, as well. Both Nietzsche and Rand see utilitarianism and Kantianism, for example, as secular implementations of the Christian value of altruism (the idea that it is morally praiseworthy to sacrifice oneself for others, especially for others in need).
Rand and Nietzsche agree that altruism is bad, but they do not agree on why. Nietzsche thinks that altruism is a sign of a sick and decaying culture that no longer values the talented individual. Unless the talented individual is able to live for his own sake, he will never achieve great things. Rand thinks that altruism is an impracticable moral ideal, one that is equally dangerous for all people: no one can consistently live by altruistic values, because their ultimate result is death. According to Rand, the deeds of great individuals merit praise and economic reward, but they do not mark out great individuals as fundamentally different in their moral or legal standing from more ordinary people. According to Rand, all people have basic rights to life and liberty. By contrast, Nietzsche seems to have thought that the naturally inferior should serve their betters as slaves.
Nietzsche and Rand share a great admiration for the ethical ideals of Ancient Greece. Nietzsche makes eclectic use of ideas from Plato, the Cynics, Aristotle, Epicurus, and the Stoics, while Rand explicitly praises both Plato and Aristotle (Rand took Aristotle to be the greatest of all philosophers). Both Nietzsche and Rand would count it as a great improvement if our society were to embrace Aristotle's ethics of human flourishing, though neither would regard this development as the ideal. Aristotle held that the most basic goal for each individual is and ought to be their own happiness or flourishing. In order to flourish, one must practice the moral virtues – temperance, honesty, courage, benevolence – that are components of self-perfection.
However, Nietzsche is opposed to the virtue that Rand identified as the most important virtue of them all: rationality. Although Nietzsche praises scientists and explorers (who are presumably rational) as superior individuals, Nietzsche ultimately sees reason as confining. He says abusive things about Socrates, because Socrates introduced philosophical reflection and thereby made mankind "absurdly rational." Nietzsche's vision of the ideal person is of one who continually creates by destroying, who is guided by primordial feelings, who is reckless and cruel to others. Nietzsche's remarks about cruelty are particularly striking. The healthy individual, uncorrupted by Christian ethics, ought to be able to torture an inferior person for fun without feeling any pang of conscience. Nietzsche writes enthusiastically about the "voluptuous pleasure" of "being allowed to vent [one's] power freely upon one who is powerless" (Genealogy of Morals, II:5).
All this emotionalism and sadism would be anathema to Rand. Her ideal person is not a "whim-worshipper," and he does not take pleasure in other people's suffering. He is rational, productive, and proud. He is a creator of life-promoting values who trades with others to mutual benefit and who lives independently and self-sufficiently – but also in harmony with others – in a rights-based democratic republic.
This difference also comes through in the historical individuals that Nietzsche singles out for praise. Rand would presumably agree with Nietzsche that Goethe and Shakespeare were great individuals; she certainly agreed with him about Dostoyevski and Mark Twain. But when Nietzsche praises Napoleon and other violent conquerors, Rand would dissent: in her view, destructive violence is inimical to human flourishing.
Sartre - Rand
Turning, now, to Sartre: Ayn Rand
and Sartre seem to agree on several important points. First, they agree that human beings have free will – that no human action is ever completely caused by what happened in the past. Second, they agree that we must create our own values and bring meaning to our own lives through a steadfast commitment to our personal projects. Third, humanity, in Sartre's terms, is "forlorn" – we are alone in the universe, there is no benevolent creator whom we can depend on or look to when deciding how we should live. Finally, both philosophers agree on a technical question in metaphysics, i.e., that consciousness is always oriented towards an independently existent object, so that it is self-evident both that one exists, and also that there is a world, outside.
Despite these similarities, Sartre and Rand are really very different philosophers. They belong to fundamentally different philosophical schools. Sartre is a phenomenologist in the tradition of Edmund Husserl, while Rand is an empirical realist in the tradition of John Locke. So, for example, Rand does not think that consciousness is a kind of nothingness, and she does not think that phenomenology (how things seem from the first-person perspective) is equally reliable "across the board" as a guide to metaphysics and ethics. In addition, their views in politics were very different: Sartre is a socialist, and he counted himself as a "fellow traveler" of the French Communist Party for several years. Finally, and perhaps most obviously, the tone of their works is very different. Sartre tends to emphasize the nauseating, disorienting, disturbing aspects of personal freedom and atheism that Rand believes are liberating. But this may not reflect any serious philosophical difference; it may just show that Sartre takes a mischievous delight in shocking the bourgeois Christians in his audience.
Kierkegaard - Rand
The similarities between Ayn Rand
and the other existentialists are slim. Like Rand, Kierkegaard was a romantic in aesthetics and a fervent individualist. However, his individualism is founded on the private and singular relation between the individual and God. Kierkegaard wrote many of his works in the voices of characters other than himself; each character is associated with a philosophical point of view (not necessarily Kierkegaard's own). Sometimes, these Kierkegaardian pseudonyms make points that agree with Rand's Objectivism
(see, for example, the letters of the Judge in Part II of Either / Or condemning the hedonistic lifestyle of Johannes the Seducer). But usually, they do not. Ultimately, Kierkegaard is an evangelical protestant who counsels belief in the absurd. This is entirely incompatible with Rand's Objectivism
, which holds that contradictions cannot be true, and that believing things simply because one wants to is a lethally dangerous habit of mind.
Heidegger - Rand
When it comes to Heidegger, we have really reached the end of the line. Heidegger's views are diametrically opposed to Rand's on almost every important philosophical question. Heidegger was opposed to reason and the scientific point of view (i.e., the view that objects have determinate, discoverable natures, independent of our beliefs and desires). He saw this scientific orientation as artificial and misleading. The more primordial point of view, according to him, is the point of view of moody, engaged activity, wherein objects have established meanings, and there is no question what things are or what they are for. Heidegger's metaphysics and epistemology end in a mystical celebration of "Being."
The list of dissimilarities goes on: Heidegger was opposed to technology, and to the technological impulse, i.e., the impulse to control or dominate nature. He held that true individuality is impossible. On his view, we all pick up our values and projects from others, and our "being" any way depends on others being willing to recognize us as being that way. He seems to have been a strict cultural relativist and ethnocentricist: one can only find meaning in the culture that one was born into. One should "be" in accordance with it and not question its mores. In politics, Heidegger was a fascist and an eager supporter of the Nazis. On all these fronts, Heidegger's views are fundamentally and totally opposed to Rand's.