Yes he did, at the age of 10 and 11, according to John Stuart Mill himself, but he never published it, and later destroyed the manuscripts, so we have no way of corroborating it.
According to his autobiography:
I have no remembrance of the time when I began to learn Greek. I have been told that it was when I was three years old.
[...]
In my eighth year I commenced learning Latin, in conjunction with a younger sister
[...]
But in my eleventh and twelfth year I occupied myself with writing what I flattered myself was something serious, and might be made fit to be published; this was no less than a history of the Roman Government, compiled (with the assistance of Hooke) from Livy and Dionysius: of which I wrote as much as would have made an octavo volume, extending to the epoch of the Licinian laws. It was in fact an account of the struggles between the patricians and plebeians, which now engrossed all the interest in my mind that I had previously felt in the mere wars and conquests of the Romans. I discussed all the constitutional points as they arose, vindicated the Agrarian law on the evidence of Livy (though quite ignorant of Niebuhr’s researches) and upheld to the best of my capacity the Roman democratic party. A few years later in my contempt of my childish efforts I destroyed all these papers, not then anticipating that I could ever have any curiosity about my first attempts at writing or reasoning. My father encouraged me in this useful amusement, though, as I think judiciously, he never asked to see what I wrote, so that I never felt that in writing it I was accountable to any one, nor had the chilling sensation of being under a critical eye.
An octavo volume is
a book size of about 6 × 9 inches (16 × 23 cm), determined by printing on sheets folded to form 8 leaves or 16 pages. Symbol: 8vo, 8°.
This doesn't give us a 100% certainty that he actually wrote it, as he never showed it anyone else, and destroyed the manuscript. And more importantly, even if it was written, we can't assess the quality of the work, whether it was a brilliant masterpiece, or a bunch of nonsense, as the most crucial part of the claim seems to be the implied competency and intellect displayed by JSM at such a young age. But we can say that prof. Sander didn't invent the claim out of thin air, but bases it on a source that is considered reliable by many people.