It's pretty obviously a conspiracy theory rooted in transmisogyny and racism.
Now, your question is a bit vague on what exactly the claim is. "chromosome abnormality" seems to indicate that Michelle Obama is intersex, while most of the rant from Jones as well as the comments from Rivers seem to indicate that Michelle Obama is trans.
Proving the absence of something isn't easy, so instead let's look at the proof given for these theories.
Origin of the theory
The first appearance of the theory seems to be from around June 2011, and it seems to have originated from Christwire.
The proof they give are:
- "medical experts" (unnamed) say Michelle Obama has broad shoulders
- the story of an unnamed "former member of the First Lady’s personal staff"
- Michelle Obama once wore a belt and danced
- a poorly photoshopped image
The proof seems ridiculous. That is because Christwire is a satirical website.
They do however create a complete backstory for Michelle. For example, they say that she was "born Michael LaVaughn Robinson", that she played football, and that she "underwent sex reassignment surgery at Johns Hopkins" in 1983.
This backstory and the fake image are important, as they are later used as proof by conspiracy theorists.
Spread of the theory
The theory doesn't really take off until 2013.
In August 2013, a website called knowthelies.com started their own theory (it seems to be independent of the Christwire article, as it doesn't reference it, nor does it include the photoshopped image).
- Michelle's look ('"Michelle" Obama's face is not handsome. It is very ugly and not feminine at all.', 'Michelle really looks like a man dressed up in drag.').
- a couple of images of Michelle Obama where her dress and pants show folds.
Now, the theory seems to spread further. In November 2013, junglesurfertv made a video showing the same images as the above article and in October beforeitsnews.com repeats the statements from knowthelies.com, but doesn't add any new proof.
Up until now, there is no proof to the theory, just some meaningless images and opinions about Michelle Obama's looks.
Michelle Obama's Children
If Michelle Obama is trans, and Barack Obama is not trans, they could not have children. Or the other way around: That they did have children is proof that Michelle Obama is not trans.
There is a conspiracy theory that Michelle's and Barack's children are actually adopted. It seems to have been started by someone named Eowyn on beforeitsnews.com.
The proof is that there are no baby pictures and no birth records. On their own blog, Eowyn had to admit that there actually are baby pictures, so their theory is not currently based on any facts.
In July 2014, Joan Rivers picks up the theory. Her proof is "We all know it".
While before only a handful of right wing fringe website posted about the theory, more and more websites talk about it after Rivers mentioned it (for example, CNN mentions it, but calls it a joke).
Now, the original article by Christwire is rediscovered and treated as a real report containing real facts, not the satire it is. Examples are atlanteanconspiracy.com, beforeitsnews.com, galacticconnection.com, and madworldnews.com.
In addition to the non-proof provided by a satirical article (the made up backstory and the obviously fake image), the proof from 2013 (Michelle's looks, and some unfortunate images), and the proof that Michelle's children are adopted (because there are few images of them), additional proof is now found:
In a speech in 2011, Barack Obama refers to someone as "Michael". "Michael" is the name that the Christwire article claims Michelle Obama had. In context, it is obvious that Obama is referring to Michael Edward Mullen, not to Michelle Obama.
There is no actual proof for the theory that Michelle Obama is trans.
The rumor started in 2013 with no proof, and was widely spread by Joan Rivers in 2014. Joan Rivers still had no proof for the claim, but with its wide spread in 2014, conspiracy theorists used a satirical article from 2011, which contains no actual facts, as proof.