What justifications are there for anarcho-capitalism?

The general justification: No man should be ruled by another man. Individual sovereignty, moral autonomy, dignity, soul, whatever you wish to call it, demands that a person refuse to be ruled. What about the “capitalist” part? There are several justifications given by various anarcho-capitalists.

Moral justifications:

The life of man qua man, man as a rational being, morally necessitates a laissez-faire economic system (Ayn Rand and objectivists).
Man must be free and uncoerced so that the man, especially his moral faculty, is allowed to evolve. (Herbert Spencer).
There is an overriding moral principle in civilised society: That no one should violate the (general moral) rights of others, that is, initiate force or threat of force. This is called the NAP — Non-Aggression Principle (Spencer, Rand, Rothbard).
Capitalism is contractual; it is what rational people implicitly agree to do when they enter society (Narveson).

Other justifications:

Libertarianism capitalism is simply what society will do more or less in many or most places in the absence of a State (David Friedman). This is a a utilitarian or “value-free” economic approach.
One cannot argue against anarcho-capitalism without implicitly agreeing to its basic assumptions (Hans-Hermann Hoppe’s argumention ethic).