A few things to keep in mind on this topic:
1) As others have mentioned no flint J&S rifles are known to exist.
2) There are no known examples of a signed Jake Hawken rifle and there are no business records from his time before working with Sam that document anything other than blacksmith work and gun repair. But his experience suggests he had the skills to build guns.
3) Sam did move to St. Louis and worked independently of Jake and Lakenan prior to Lakenan's death in 1825 and Jake and Sam's merger that year.
4) I am aware of one brass-mounted, script-signed S. Hawken long rifle of Maryland styling that was found in the St. Louis area. The gun is original percussion and is thought by the owner (one of the most knowledgeable Hawken scholars in the country) to come from the 1822-25 period in STL.
5) During the 1820s and early 30s, the market was pretty well flooded with inexpensive, well-made flintlock rifles from Lancaster, Philly, Hagerstown, and Europe. STL was a major crossroads for trade and newspaper adds of the period suggest there was no shortage of quality arms available.
The following are my conclusions from the points above.
Flint J&S rifles were likely made, but were likely uncommon and the lack of surviving pieces and the numbers of surviving percussion pieces both support that. In order to build a clientele and a reputation in STL, Jake and Sam likely had to differentiate themselves from the masses by offering something better than what was commonly available. There are several existing pieces from theirs early years together that support this. They produced fine percussion guns for the important STL families and fur trade bourgeois. The first records of guns showing up for sale at rendezvous are in 1834, and it is just a handful of guns nine years after Jake and Sam went into business together. I believe the evidence shows that Hawkens were top drawer guns throughout the 1830's and 40's, and men laying down that kind of money probably went most commonly for the most up to date technology.
This does not in any way say that no flintlocks were built. The oddball Smithsonian gun Dan mentions above proves that at least one guy wanted Sam to build a flintlock after James death. It just says that people have been looking for a flint J&S since Horace Kephardt started repopularizing Hawkens in the 1930s and none have been found. If a guy wants a flint J&S, more power to him. However it was probably an oddball gun back then.
Your mileage may vary.
Sean