“The feature of the game was a fine running catch by Force in the second, and another magnificent running fly-catch by Burroughs in the third inning…”
“Base Ball – Olympics of this City versus the Olympics of Baltimore” Washington (DC) Evening Star, 11 August 1871, Page 4.
Henry was one of the members of the original Washington baseball teams, playing a number of different positions for the Olympics in 1871 and 1872 when Washington had an entry in the National Association. In 12 games in 1871, he contributed 15 hits – six for extra bases, including a homer – and then appeared in two games the following season, collecting a single hit in his seven official at bats (and a walk). From what I can tell, he played with the Olympics in 1870 prior to that team joining the National Association. It was a professional team, with players being hired from a variety of different cities in addition to the local ball players.

Burroughs with his Olympic Teammates. Burroughs is right in front (#9), with Dave Force (#7) to the left, Asa Brainard (#8) behind him to the left, and Doug Allison (#10) behind to the right.
Arriving 03 February 1845, Burroughs was the fifth of six children born to Joseph and Hannah Palmer (Searles) Burroughs in Essex, New Jersey. Joseph was a fur blower and made hats while Hannah was busy raising children. It looks like Henry worked for his father when not playing ball. He stayed in the D.C. area after his playing days taking a clerk position an government auditor office. He returned home to Essex and married Leila A. Van Husen in March 1876. Unfortunately, he fell ill two years later and died 31 March 1878. Leila never remarried – she died in New Jersey in 1939.
Sources:
1850 US Census
1865 Newark City Directory
FindAGrave.com (Henry)
FindAGrave.com (Leila)
1873 US Register of Civil, Military, and Naval Service
Baseball-Reference.com
“Base Ball.”, National Republican, 18 June 1870, Page 4.
“Base Ball – Olympics of this City versus the Olympics of Baltimore” Washington (DC) Evening Star, 11 August 1871, Page 4.
The Miriam and Ira D. Wallach Division of Art, Prints and Photographs: Photography Collection, The New York Public Library. “Olympics of Washington, D.C., A. J. Leonald, l. f., G. W. Hall, c. f., H. W. Berthrong, r. f. , F. A. Waterman, 3b., C. J. Sweasy, 2b., E. Mill, 1b., D. W. Force, s. s., Asa Brainard, p., H. F. Borroughs, D. L. Allison, c., J. W.” New York Public Library Digital Collections. Accessed August 19, 2020. http://digitalcollections.nypl.org/items/510d47d9-c2c4-a3d9-e040-e00a18064a99