That Unz page promotes a theory that the first case(s) appeared in Virginia in the summer of 2019. So that should give some idea where it's headed with that premise. It also claims that (all?) CDC discussions of Covid-19 are classified as "Top Secret". (I'd be surprised if any of them are.)
The CDC might themselves not actually know who the real "Patient Zero" was in the US, so the question may be moot in that sense. It's also not a very meaningful question because--as noted in comments--there probably were multiple introductions of Covid-19 to the US from abroad, so which one was the absolute first on timeline might not matter much either.
The initial US cases that have been discussed in the scientific literature do give the usual (anonymized) data that's given in such reports, e.g.
First Case of 2019 Novel Coronavirus in the United States [...]
On January 19, 2020, a 35-year-old man presented to an urgent care clinic in Snohomish County, Washington, with a 4-day history of cough and subjective fever. [...] He disclosed that he had returned to Washington State on January 15 after traveling to visit family in Wuhan, China. The patient stated that he had seen a health alert from the U.S. Centers for Disease Control and Prevention (CDC) about the novel coronavirus outbreak in China and, because of his symptoms and recent travel, decided to see a health care provider.
And as further detailed in the paper, PCR tests for him turned out negative for basically every respiratory pathogen but the new coronavirus.
Note that even for a (retroactively diagnosed) French "first" case, whose name is well known (because he didn't shy the press--he gave TV interviews etc.), the actual scientific/case report published in a medical journal omits his name.
As for the relevant US laws, the CDC notes for example that
Federal law, upheld by the Fifth Circuit Court of Appeals**, prohibits NCHS from releasing personal information to anyone without consent – no matter who they are and no matter how carefully they say they will take care of the information.
The restriction on who gets to see personal information extends from the highest levels of our government (we can deny the President and any member of Congress access to confidential information as well as Immigration, Justice, and IRS officials) to the kinds of inquiries we are all increasingly worried about (market research firms, insurance companies, employers).
These are not just promises. NCHS has been collecting health statistics since 1957 and in all that time we have not released any confidential information to anyone not entitled to have it. That’s not because there have been no requests for personal information or because no special precautions are needed to prevent a disclosure. It’s because we take the law and our ethical obligations seriously enough to be constantly concerned.