I think I can contribute some points even though I have to admit limited knowledge about Nietzsches philosophy.
What I do know from biographical information ("Godess of the market" by Jennifer Burns) is that she was familiar and quite sympathetic to Nietzsche in her younger years.
From that biography I also have the following quotation:
[...]as when she wondered, if perhaps "the rational faculty is the dominant
characteristic of the better species, the Superman."
(The text before the double quotes is by Burns and she gives the sources as Journals, 291, 281 and 285.)
As someone who is extremely appreciative of Rand and at least superficially familiar with Nietzsche I venture to guess that they indeed both understood by that the same (and to me rather obvious) idea: That only very few people achieve great things while a majority likes telenovelas.
A more well-known connection to Nietzsche can be found in the foreword of "The Fountainhead", where Rand admits to have considered putting the following quotation in her novel:
It is not the works, but the BELIEF which is here decisive and
determines the order of rank--to employ once more an old religious
formula with a new and deeper meaning--it is some fundamental
certainty which a noble soul has about itself, something which is not
to be sought, is not to be found, and perhaps, also, is not to be
lost.--THE NOBLE SOUL HAS REVERENCE FOR ITSELF.
(From beyond Good and Evil by Nietzsche.)
Rand tells us that she decided against using the quotation because it "proclaims psychological determinism" - something she abhorred.
I believe that the quotation might be a key commonality between Nietzsche and Rand in contrast to most philosophers and intellectuals: To give one's own life importance rather than sacrifice it to something or someone else.
In the foreword she goes on to say about the quotation:
This view of man has rarely been expressed in human history. Today, it
is virtually non-existent. Yet this is the view with which—in various
degrees of longing, wistfulness, passion and agonized confusion—the
best of mankind’s youth start out in life. It is not even a view, for
most of them, but a foggy, groping, undefined sense made of raw pain
and incommunicable happiness. It is a sense of enormous expectation,
the sense that one’s life is important, that great achievements are
within one’s capacity, and that great things lie ahead.
This view might be one of the crucial intersection of Nietzsche with Rand and thus give the impression of her being a blend of Nietzsche with something else.
These points of contact nonwithstanding, the claim of Rand's philosophy being a blend of Nietzsche with American individualism is wrong. Since American individualism isn't an actual philosophy, it suffices to show philosophical work that has no precedence in Nietzsche.
I've picked three articles that together spread over a range of fields in philosophy and should be comprehensible and intriguing. They should show that there are numerous gems of ideas in Rand's philosophical work - gems that (to my knowledge) are not of Nietzschean origin.
I also made sure to pick unusual ones (as opposed to writings dealing with selfishness, capitalism and reason, which are the usual suspects when it comes to Rand).
- 'The anatomy of compromise' in 'Capitalism: The Unknown Ideal'
- 'Art and moral treason' in 'The Romantic Manifesto'
- 'The establishing of an establishment' in 'Philosophy, who needs it?'
I strongly recommend reading these articles.