I have no idea from where these Objectivists get the idea that Greenspan is (or was) a follower of Rand. He has never said anything like that. Sure, he talks about his admiration for Rand's philosophy, but he does not see himself as her follower. In fact, he disagrees with Rand on a number of issues.
In his book The Age of Turbulence: Adventures in a New World, Greenspan talks about why he disagrees with Rand’s theory of taxation. Here’s an excerpt:
According to Objectivist percepts, taxation was immoral because it allowed for government appropriation of private property by force. Yet if taxation was wrong, how could you reliably finance the essential unctions of government, including the protection of individuals’ rights through police power? The Randian answer, that those who rationally saw the need for government would contribute voluntarily, was inadequate. People have free will; suppose they refused?
I still found the broader philosophy of unfettered market competition compelling, as I do to this day, but I reluctantly began to realize that if there were qualifications to my intellectual edifice, I couldn't argue that others should readily accept it. (Page 52)In the book, he also says that Rand used to refer him as "the Undertaker," partially because his manner was so serious and partially because he always wore a dark suit and tie. "Over the next few weeks, I later learned, she would ask people, "Well, has the Undertaker decided he exists yet?" (Page 41)