What makes people irrational thinkers?

[LadyForCamus]

I think that by "account for its own existence" he means that there is nothing within nature which explains why it exists to begin with.

Sure, but as Polymath already pointed out, this begs the question, as it assumes there is a “why” to existence, and it assumes that the facts of physical reality are not a sufficient explanation. These assumptions need to be demonstrated, otherwise he’s simply sneaking his conclusion into his argument. 

Science describes the causes and effects within nature, but not the cause of nature itself.

Hart, himself recognizes the logical mistake of assuming that the rules which apply within our physical reality don’t necessary apply to physical reality as a whole. Therefore it would be a composition fallacy to assume that because causality exists within the universe/physical reality there must necessarily be some cause of it. Again, he needs to show that “stuff exists necessarily” can’t be a possibility. He doesn’t get to just assume it for the sake of his argument. 

To say that nature causes itself, you'd need a strange argument: 

 ~ First no part of nature existed.
 ~ Then a part of nature caused nature to exist.

Or, nature exists necessarily. How can no part of nature exist? How can “nothing” be an existing state of affairs? What is a state where non-existence exists? This is the fundamental problem (IMHO) with theistic “something rather than nothing” arguments. How can nothing have been a possible alternative to something?

1. Hart hasn’t shown here that physical reality can’t be causeless and necessary. 

He certainly hasn't in the quote above. Does he address the issue elsewhere in the book?

No, you’re right. He doesn’t show it. He attempts to reason to it, but his reasoning is fallacious. I don’t know. I have two little ones at home with me and I cannot sit down and read this entire book. From what I understand, you have, so if you’d like to reference his supporting evidence here with a page number I’d be happy to take a look at it.

2. If nature cannot account for itself by the fact of its very existence, then this seems to also be a problem for God’s existence accounting for itself.

This, too, isn't addressed in the brief quote you give, but is something for which there are other arguments. Very roughly, the idea is that things which can't be self-caused are contingent, and therefore must rely on something necessary.

Again, until he can show that physical reality itself can’t exist necessarily, his argument is begging the question.