The tricky part of your question is when you ask "Satsuma rebellion and presumably the Boshin War".

Foreign involvement in the Boshin War was rather different than foreign involvement in the Satsuma Rebellion. American involvement perhaps even more so.

By the time of the Satsuma Rebellion, the Japanese Military was relatively uniform. Not when compared to European nations, but when compared to the Boshin War. When Akiyama Yoshifuru, the father of Japanese cavalry, was sent to France, he wrote back to Japan about the bias the Army had for anything Prussian. This was because Prussia had won the Franco-Prussian War and thus they saw it as superior. The Japanese Army had Prussian weapons, Prussian instructors, Prussian doctrines, Prussian traditions, etc. Of course, there were exceptions.

The Japanese Navy was in a similar position with the Royal Navy. The navy followed British doctrine and traditions, and had British instructors. While they did have Krupp make the cannons on their ships, the ships were mostly designed in Britain. The only exceptions to these were a couple of French ships to test the Jeune Ecole doctrine when it was birthed.

This was not only because they saw those nations as the best in their fields, but also because having uniform systems is essential for maintaining good logistics.

The Boshin War, however was very different. The sheer variety of weapon types used was immense. Tanegashima muskets and katanas were used along chassepots and snider enfields. This is because, instead of having a national army, the war was fought between individual clans who obtained whatever they could get from foreign powers.

For example, the Imperial Navy was not a unified national force, but the combined navies of Satsuma, Tosa and their allies. Even within the same clan, there were many different ships because they bought whatever was available. The result was that the Imperial Navy was a chaotic mess of ships from multiple nations, going at different speeds and firing different ammunition.

In this way, the Boshin War saw both Imperial and Shogunal Forces using a wide variety of weapons from a variety of nations, including the United States.

While the US was neutral in the conflict, it was not shy in selling weapons: especially surplus from their own civil war.

This includes repeating rifles, revolvers and gatling guns. While technically originally French in design, the US also sold the Confederate Ironclad CSS Stonewall to the Imperial faction. Eventually both sides would favor the chassepot and snider enfield, but there was no absence of American weaponry.

Thus US weaponry was much more prevalent in the Boshin War than the Satsuma Rebellion.

As far as Foreign Advisors go, while the roles they were given were not the same as in the movie (in fact the same can be said for all nations), Japan did request foreign advisers from the States.

However, most of these advisors were not military, but civilian. American advisors were highly sought in the field of agriculture and the near entirety of the Island of Hokkaido (which was newly settled at the time) based their farms off of American farms. So America did play a prominent role in the modernization of Japan, and did send foreign advisors, just not the ones depicted in the movie. Again, the same can said for all nations mentioned in the movie.

That isn't to say that America didn't shape anything military in Imperial Japan. The first Naval Academy, and their wargames, were heavily based on the ones in America. Furthermore, although not directly involved, the works of Alfred Thayer Mahan was instrumental in developing Japanese Naval Doctrine.

I've realized that, past the already well-answered direct response to your question about American involvement, there's one more thing to mention.

Algren in The Last Samurai is often thought to be based off Jules Brunet, the French military advisor to the shogunate, who in the Boshin War defied France's agreement to neutrality, and along with a few other French officers, joined the side of the former shogunate retainers. Brunet fought the war to the bitter end, organizing the short-lived "Republic of Ezo" (present-day Hokkaido was then called Ezo) with other shogunate loyalists like shogunate naval hero Enomoto Takeaki, Hijikata Toshizo of the (in)famous Shinsengumi, and the somewhat mediocre general Ootori Keisuke.

These final hold-outs against the new Imperial state have been romanticized as the loyal old-fashioned samurai who held to the code of Bushido to the bitter end. (Which is not really fair to some of these quite modernized rational fighters.) The events and Japanese characters of The Last Samurai are more based off the Satsuma Rebellion, but it makes sense that Algren is inspired by Brunet who is associated with this final stand of honour.