There is no doubt that Green & Elizabeth divorced.  The fact that, in the 1870 census, neither of the two children born by 1870 are living with Elizabeth suggests she lost custody of them or that Greenberry deserted her leaving her no choice but to place them where she could.  Also, the fact that, in the 1870 census, Minnie BAIN (æ 5) is living with Green's widowed mother, Rachel (LARRISON) BEAN/BAIN, but Charles BAIN (æ 3) is living in a public orphanage, suggests that only Millie was Greenberry's biological child.  Green and Elizabeth had clearly separated by the 1870 census because, though I can't find Greenberry at all, Elizabeth (still surnamed BAIN) is living with Elijah HAMBLETON (a.k.a., Elisha HAMILTON); and, as mentioned, neither of her children is with her.  In the 1880 census, Lillie and Harry are with Elizabeth, but are surnamed HAMILTON.  That Elizabeth is still surnamed BAIN in 1880 indicates she did not marry Elisha, though he must have acknowledge his fatherhood of Lillie and Harry for them to have legally used his surname.  Son George carries the BAIN surname apparently because he was born illegitimately, after the liaison with Elisha HAMILTON had ended — and after Greenberry had remarried to Frances (TOLL) HANKISON. 

All of the above is supposition based on circumstantial evidence.  County Court documents should be sought for records of the divorce, guardianship of the children, and any bastardy claims by Elizabeth. 

Y-DNA testing of a male patrilineal descendant of Harry should determine the identity of his actual father.  Charles had no surviving children, so Y-DNA testing cannot be used to prove who his father was.

1.  LDS.  Family Search: Internet Genealogy Service:  IGI - International Genealogical Index (online at FamilySearch.org).

Hezekiah HYATT Spouse: Sarah HARTMAN Marriage: 04 Dec 1830, Guernsey, Ohio Source: Marriage Records, 1810-1951. Ohio. Probate Court (Guernsey County)

2.  1830 Census Index/Images (online at Ancestry.com):  Vincen BEAN (æ 30-40) is living in Blue Rock Twp., Muskingum Co., OH (p. 242B).  Also in Blue Rock Township are James LARACEN (æ 50-60) and William ROCKEY (æ 40-50).

3.  1840 Census Index/Images (online at Ancestry.com):  Vincent BEAR / BEAN (æ 40-50) is living in Harrison Twp., Muskingum Co., OH (p. 357B); he is listed four lines below Henry KOON (æ 20-30), who is one line below Wm. ROCKEY (æ 40-50).  There are three Hezekiah HYATTs indexed in Ohio, all of them in Muskingum County.

4.  1850 Census Every-Name-Index/Images (online at Ancestry.com):  Greenberry / Greenbury BANE (æ 18, b. OH) is living in Brush Creek Twp., Muskingum Co., OH (p. 65A), with his widowed mother, Rachel BANE (æ 50, b. PA), plus siblings.

5.  1860 Census Every-Name-Index/Images (online at Ancestry.com):  Green BAIN (æ 24, b. OH) is living in Brush Creek Twp., Muskingum Co., OH (p. 400A), with his widowed mother, Rachel BAIN (æ 59, b. OH), plus sibling.

6a.  1870 Census Every-Name-Index/Images (online at Ancestry.com):  can't find Greenberry, and I really looked.

6b.  1870 Census Every-Name-Index/Images (online at Ancestry.com, Image #16 of 32):  Washington Twp., Muskingum Co., OH, Roll M593_1251, p. 425B, PN 16, 121/120, enumerated 13 Jun 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):

Hambleton Elijah	62	M	W	Farmer	1200	120	East Virginia
Bain Elizabeth	30	F	W	House Keeper			Ohio

6c.  1870 Census Every-Name-Index/Images (online at Ancestry.com, Image #6 of 29):  Duncan Falls P.O., Taylorsville, Harrison Twp., Muskingum Co., OH, Roll M593_1250, p. 110B, PN 6, 47/47, enumerated 4 Jun 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):

Bean Rachel	69	F	W	Keeping house	250	140	Pennsylvania
____ Minnie	5	F	W	Attending School			Ohio

Rachel is Minnie's paternal grandmother.  Minnie's brother, Charles BAIN, is living in an orphanage in Zanesville (see next source).

6d.  1870 Census Every-Name-Index/Images (online at Ancestry.com, Image #161 of 254 — indexed "Barri"):  Orphans Home, Zanesville, Muskingum Co., OH, Roll M593_1251, p. 534A, PN 29, 224/224, enumerated 27 Jul 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):

Bain Charles

3

M

W

At School

Ohio

One of 22 children in the orphanage.  The fact that Charles ended up in an orphanage, while his older sister, Minnie, was taken in by our subject's widowed mother, suggests that only Minnie was Greenberry's biological child.

7a.  1880 Census Every-Name-Index/Images (online at Ancestry.com, Image #9 of 11):  Taylorsville, Muskingum Co., OH, Roll T9_1054, p. 93B, PN 26, SD 7, ED 157, enumerated 16/17 Jun 1880, official enumeration date 1 Jun 1880 (extracted by Diana Gale Matthiesen):

1880:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).
1	2	3	4	5	6	8	10	13	24	25	26
224	229	Bain Green	W	M	47		/	Grocer	Ohio
		"  Frances	W	F	41	Wife		Keeping house	"	Ohio	Ohio

7b.  1880 Census Every-Name-Index/Images (online at Ancestry.com, Image #28 of 31):  Girls' Industrial Home, Concord Twp., Delaware Co., OH, Roll T9-1012, p. 275D, PN 31, SD 5, ED 101, enumerated 19 Jun 1880, official enumeration date 1 Jun 1880 (extracted by Diana Gale Matthiesen):

1880:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).
3	4	5	6	8	9	13	21	24	25	26
Bain Minnie	W	F	16	Pupil	/	Domestic Servant and Scholar	/	OH	US	US

One of 259 girls in the "school."  Minnie's brother, Charles BAIN, is still living in an orphanage in Zanesville.

8.  1890 Census:  the 1890 Census Population Schedules were destroyed.

9.  1900

10.  1910

11.  Woodlawn Cemetery, Zanesville, Muskingum County, Ohio (online at Find-A-Grave.com):

Green B. Bain Birth: Jul 3, 1833 Death: Feb. 26, 1893 Inscription: aged 59y 7m 23d Note: on stone with George W. Hankison Submitted by Robert Weller

12.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com).  A file there includes this obituary:

Obit courtesy of the Zanesville Times Recorder [4/27/1919] BAIN FUNERAL HERE ON WEDNESDAY NOON The body of the late Mrs. Frances Bain, aged 81, a former resident of this city who died Sunday at the home of her son, G.W. Hankinson, in Cleveland, will arrive in this city over the Wheeling & Lake Erie railroad Tuesday evening and be taken to the Hearing morgue.  The funeral will be held there Wednesday afternoon and interment will be made in Woodlawn Cemetery. Besides the son with whom she made her home, she is survived by four sisters, three brothers and three grandsons.

13.  Email from Carol Berry.