I've written already quite a lot on the Burnham brothers, two of whom are my ancestors (Robert and Thomas). I've discussed how they were not on the ship Angel Gabriel. It then resulted in a discussion of when they did arrive in New England. So you can review those postings which remain up to date in their content.
1. Robert Burnham, born ca. 1624 (deposed as aged 55 in 1679), died at Ipswich, Massachusetts while visiting on 12 June 1691. Robert married about 1646 probably in Boston, Frances (---) who survived him. Robert is in Boston first, then removes to Oyster River (present day Durham) New Hampshire. He is called kinsman in the will of Robert Andrews of Ipswich, Massachusetts. He had six children: Robert, Samuel, Elizabeth, Jeremiah, Sarah, and Robert.
2. Samuel Burnham, born ca. 1649 probably in Boston, died 1695-96 in Oyster River, New Hampshire. Samuel married about 1675 Mary (---) who was alive in 1689. Samuel had three children: James, Frances, and Mary.
3. Mary Burnham, born abt 1688 at Oyster River (Durham), died before 10 July 1777 at Rochester, New Hampshire. She married about 1716, John Allen, son of Charles and Susannah (Huggins) Allen.
The best source for this family is the Ancestry of Charles Stinson Pillsbury & John Sargent Pillsbury by Mary Lovering Holman (2 vols., 1938). However, they also appear in Genealogical Dictionary of ME and NH by Sybil Noyes, Charles Thornton Libby, and Walter Goodwin Davis (Portland, 1939, reprinted Baltimore, 1988), p. 121 and History of the Town of Durham, New Hampshire (Oyster River Plantation) with Genealogical Notes by Everett S. Stackpole & Lucien Thompson (2 vols., Rumford Press, Concord, NH, 1913, reprint, Somersworth, NH 1973), p. II:40. Just remember to ignore the Angel Gabriel stuff.