John Greene was born ca. 1606 near London, England, probably Enfield, England. He was variously called John Greene the Younger, John of London, John of Wickford and John of Quidnessett.
At around age 36 he married a young widow named Joan Beggarly.
As a young man, in 1635-1636 he migrated to the New World, sailing on the ship, Matthew landing first at St. Christopher, British West Indies. Finding the people there a "Godless set", he sailed to Massachusetts, the great Puritan settlement. 
Differing with the religious authorities there, he accompanied Richard Smith to Narragansett Bay where the latter had a trading post at Quidnessett, or Aquidneset (nka Rhode Island) by 1637. The two established a flourishing trade with the Indians. For some years, Smith and Greene were the only white settlers at Quidnessett. Roger Williams and a Mr. Wilcox purchased land there in 1643 or 1644, and Williams sold his holdings to Richard Smith in 1651. On June 11, 1659 the Indian Sachem, Coquinaquant, sold the entire region of Quidnessett, which had belonged to the Narragansett Indians, to a land company headed by Major Humphrey Atherton. There were few Rhode Islanders in the company but Richard Smith was among them, the rest being Boston and Connecticut speculators.
On 24 March 1682 he divided Land among certain sons, and in
1695 he died at Quidnessett, Rhode Island.
There were two other John Greenes who arrived in Rhode island around the same time. He is not to be confused with John Green, surgeon of Warwick, RI who arrive in Rhode Island at about the same time and whose wife was also Joan but Tattersall.
Although there are two published sources from the early 1900s claiming a direct pedigree from John Greene through Robert Greene to royalty in England, the supposed source documents for this pedigree (wills, deeds, etc.) have never been located and these sources have been discounted by genealogists. Genealogists generally consider these lines to be fabricated. La Mance does not publish any of the documentation she claims to have received while building this pedigree and Rixford uses La Mance’s book as her source.
The two sources which have been invalidated by genealogists and SHOULD NOT BE USED to source this pedigree are:
La Mance, Lora S. The Green Family and all its Branches from A.D. 861 to A.D. 1904. Floral Park, NY: Mayflower Pub. Co., 1904. Page 33.
Rixford, Elizabeth M. Leach. Families Directly Descended from all the Royal Families in Europe (495 to 1932) and Mayflower Descendants. Genealogical Publishing Co., 1932. Page 74.
Children list from Austin: John of Kings Town and wife Joan
John b. 6 June 1651; d. 1729; m. Abigail
James b. 1655; d. 1728; married (1) Elizabeth; m. (2) Ann
Daniel d. 1730; m Rebecca Barrow
Edward m. Mary Tibbitts
Benjamin d. 1719; m. Humility Coggeshall
Children list from the Settlers of the Beekman Patent.
Edward b. c. 1643; m. Mary Tibbetts
Daniel b. c 1647; died 1730; m. Rebecca Barrow (perhaps a 2nd m.)
Henry b. c 1648-50; m. Sarah d/o John Greenman. Went to Monmouth, N.J. died there c 1694
John b. 16 Jun 1651; m. Abigail
Robert b. 1653
James b. 1655; d. 1728; m. ?Elizabeth Jenkins; m. (2) Ann
Dutchess County, NY: The Settlers of the Beekman Patent (Online database: AmericanAncestors.org, New England Historic Genealogical Society, 2016), (Orig. Pub. by Frank J. Doherty, Pleasant Valley, NY. Frank J. Doherty, The Settlers of the Beekman Patent, Dutchess County, New York: An Historical and Genealogical Study of All the 18th Century Settlers in the Patent, ten volumes. 1990–2003). https://www.americanancestors.org/DB409/i/12175/752/0