FYI regarding John C. Hamilton (ca 1770-1812). He was married to Martha Greer, the d/o James Greer who appeared on the 1790 Laurens Co., SC census. John C. Hamilton may be the son the a William Hamilton who was on the 1790 Newberry Co., SC census. This is the same area and time that Dr. John H. Bush was in those 2 counties of SC. I don't know whether there's a clue hidden there or not.
John C. Hamilton was the security on the will of a Abner Bush in 1787 in Sumner County, TN. Abner Bush's 14 year old son, Abner chose John C. Hamilton as his gaurdian after his father's death. That would indicate that Hamilton and Abner Bush were likely close. Interestingly - Dr. JOhn H> Bush provided security for John C. Hamilton as Abner's guardian.No one has yet to find a link between John H. Bush and Abner Bush but one might well exist. It seems very plausible.
WE have never actually proven Dr. JOhn H. Bush's fathers identity. There may be clues in the naming pattern he chose for his own sons. 1. William B. Bush 2. George Bush 3. John B. Bush (likely B stands for Blalock, Bush's grandfather in law) 4. Alexander Graham BUsh (named for an Alexander Graham with who Bush was associated in Sumner Co. when young Alex was born) 5. Palemon Cage Bush ( named for others in Sumner Co., TN when he was born - Major William Cage 6. Reuben Rowland Bush (named for Bush's father in law, Reuben Rowland)