There are two Van Fleets mentioned in the book, Garrett and John.
Garrett is included in a list of citizens who participated in the Civil War in the 77th Regiment, company F. He enlisted on Feb. 11, 1864 and mustered out with the company. His regiment was recruited/organized in Marietta, OH, before proceeding to Camp Dennison and then on to Paducah, KY to report to General Sherman. They ended up deep in the south in Mississippi and under Sherman, participated in the Siege of Corinth, which resulted in a Union victory. The regiment was disbanded shortly before the end of the war in April 1865 in Columbus, numbering 17 officers and 348 soldiers.
John van Fleet was also in the same regiment. He enlisted on Dec. 13, 1861 and he is listed as missing in action. There's no mention of when he went missing.
I hope some of that info was useful!