Henry was the oldest child of Daniel and Susannah as far as I know.

His tombstone give DOB as 14 Aug 1806
His Obituary says he came to marshall county, IN 1861

Enumerated 1810 Census as of August 6 1810 in Allegheny, Somerset County, Pennsylvania, USA
Daniel Burket
Males under 10 = 2 (Henry & George)
Males 16 to 25 = 1 (Daniel)
Females 16 to 25 = 1 (Susannah Mull

Enumerated 1820 Census as of August 7 1820 in , Perry County, Ohio, USA
Jackson Twp
Daniel "Burguid"
Males under 10 = 3 (George, Daniel, Peter)
Males 10 to 15 = 1 (Henry)
Males 26 to 44 = 1 (Daniel Burket)
Females under 10 = 3 (Elizabeth, Mary "Polly", Susan)
Females 16 to 26 = 1 (Susannah Mull)

Married: 8 May 1828 in , Perry County, Ohio, USA to Catharine Houts
Note: John Vannatta J.P.
Marriage License: 3 Apr 1828 in , Perry County, Ohio, USA

Enumerated 1830 Census as of August 7 1830 in , Perry County, Ohio, USA
Jackson Twp
Males 20 to 29 = 1 (Henry Burket)
Females under 5 = 1 (Margaret "Rebecca")
Females 10 to 14 = 1 (?)
Females 20 to 29 = 1 (Catharine Houts)

Obituary Rochester, Fulton County, Indiana, USA May 1892
Another old land mark has gone in the death of Henry Burkett, who died at the home of his son Ephraim in Richland Township, Fulton County, last Sunday morning, May 1st, at the age of 85 Years, 8 months and 17 days. His funeral took place at the Richland M.E. Church and the burial at Burr Oak, this county, Monday. Mr. Burkett came to Marshall county in 1861, and opened and lived on a large farm in Union Township, this county, untill about four Years ago, when his wife died, at the age of 84 Years. For 50 Years he was a member of the Evangelical Church. he raised a large family of 9 children, seven of whom survive him, namely Rebecca Overmyer, age 63; Levi, aged 61; Mary Overmyer, aged 59; George H., aged 57; Susannah Overmyer, aged 54; Ephraim, aged 52; and Lucinda Working,the youngest. Some 75 grand children, and about the same number of great grandchildren honor his memory. One striking circumstance in relation to this well known family is that all the sons married Overmyers, women, and all the daughters married Overmyers men except the youngest, Lucinda, whose name is Working. Two well known and highly respected families are thereby pretty thououghly related.