I've been looking at the members of the Cyclone Club and noticed that several of the members were either in the military in 1860 or would go on to have prominent military careers.
Merritt Griswold was a Captain with the 3rd regiment of the United States Reserve Corp. Orville Matthews was a graduate of the Naval Academy, Class of 1855. Joseph Fullerton was practicing law in St. Louis in 1860 but would serve as an officer in the Civil War and rise to the rank of General. Edward Bredell was an officer in the Confederate army and served with Mosby's Rangers. Charles Kearny was the son of General Stephen Watts Kearny. Fred Benton would serve as a Captain under General Custer.
One would have to assume it's common for teams of this era to have a high percentage of members who would serve in the Civil War. But I think the Cyclones may be a bit different. Matthews was an active duty naval officer in 1860. It's likely that Kearny was an active duty army officer and Griswold was already a member of Reserve Corp. This leads me to speculate that the Cyclone Club may have been formed by some of the military men in St. Louis and their friends (heavy emphasis on speculate).
Add to this the fact that Jeremiah Fruin was in the army prior to the Civil War and was stationed in St. Louis in 1861 and one can begin to see a pattern. The influence of military men who came to St. Louis prior to and during the Civil War on the development of baseball in the city seems to be rather prominent and demands more research.
Note: The picture at the top of this post is of the cemetary at Jefferson Barracks around the turn of the century.