Another site lists a Henry Bodine, born 1832, as one of Joseph's children. I wonder if this might be a John Henry Bodine.

From Ronny Bodine:

Obituary, Mount Morris Union Constitution (Mount Morris, N.Y.) of Thurs of 22 June 1865.
DEATH of J. H. BODINE--We regret to announce the death of Mr. John H. Bodine of this village, which occurred on Friday last, after a short illness. Mr. B. was a member of the firm of J. H. Bodine & Co., manufacturers of Agricultural Implements, Machinists, &c. He was a man of untiring industry, of a genial nature, uniformly upright, kind and generous to a fault, and sincerely esteemed as a friend and citizen. Our community share with his relatives in this unexpected bereavement. He was buried on Sunday with Masonic honors a large concourse in attendance.