I found information on two individuals whom served as officers for the 6th Massachusetts Regiment also known as the 4th Continental Regiment. Which fought at Breeds Hill (Bunker Hill). Please see Information below.
Richard Buckmaster- Served as lieutenant, 2d lieutenant and captain from December 1775 until his death in November 1779.
William Buckmaster- Served as lieutenant-colonel from December 1776 until his death on November 1776.
I found the same information as Elliot Schneider....Could not locate a Benjamin Buckmaster at the Battle of Breed's Hill (Bunker Hill) Sorry
I did find a Benjamin Buckmaster who served in the War of 1812. on familysearch.org
familysearch.org is a free site...you just need to register to search
You might be able to get more information on him at Fold3.com...it is a pay site, but they do give a free trial.
Dear Mr. Buckmaster,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located the Compiled Service Records of Soldiers Who Served in the American Army During the Revolutionary War in the War Department Collection of Revolutionary War Records (Record Group 93). Portions of this series have been digitized and are available online through the Catalog. We searched for persons named Buckmaster and located 4 results, none of which have the first name of Benjamin. For more information about these records and other records in NARA custody concerning the American Revolution, please email the National Archives at Washington, DC - Textual Reference (RDT1) at Archives1reference@nara.gov.
Due to the COVID-19 pandemic and pursuant to guidance received from the Office of Management and Budget (OMB), NARA has adjusted its normal operations to balance the need of completing its mission-critical work while also adhering to the recommended social distancing for the safety of NARA staff. As a result of this re-prioritization of activities, you may experience a delay in receiving an initial acknowledgement as well as a substantive response to your reference request from RDT1. We apologize for this inconvenience and appreciate your understanding and patience.
The service records are available in their entirety via Fold3 as Revolutionary War Service Records. You may search for free, however viewing the documents turned up by search results requires a paid subscription to the website. Some libraries provide free access to Fold3 to their patrons. We searched these records on Fold3 and located the same 4 results available on the National Archives Catalog.
Many books have been published about the battle, and some may list the senior officers present. You may wish to contact the Massachusetts Historical Society, which has a number of collections related to the American Revolution, particularly Massachusetts units. Other institutions which might have useful information include the Massachusetts Archives, the State Library of Massachusetts, the Battle of Bunker Hill Museum, the American Revolution Museum at Yorktown, the Museum of the American Revolution, the Fred W. Smith National Library for the Study of George Washington, and the Daughters of the American Revolution.
We hope this is helpful. Best of luck with your family research!