|Moses, named as a cowkeeper|
Moses was granted 7 acres of land for a house lot in 1640, making him one of the early settlers of Hampton. He was not married at the time. His name was recorded by Edward Colcord on his list of the "First-Comers" to Hampton. He is listed a young unmarried man with a lot. Although, if he were born around 1594 then he was 46, not exactly young.  He was given the job of herdsman at a January 1641 town meeting. In 1645 he was granted two shares of the cow common.  He was given various other land grants over the years.
All New England men had to perform various duties in support of their town. Some did more than others. Moses held the minor post of cow keeper in his early days in Hampton. It was not until 1662, when he was almost seventy years old, did he hold on the position of Prudential Man. One of the men charged with the running of the town for the year. In 1679 he was on a list of men who could vote in the General Assembly. In 1681 he was officially freed from the duties of the train band. This was normally done around the age of 60 or so. Either Moses was an exceptionally fit man for his advanced age or he was not born as early as 1594.
marriage and family
The date and circumstances of Moses' marriage are unknown to us. He probably have a limited pool of potential wives to choose from. Daughters, sisters and widow's of Hampton men were his most likely choice. The date of birth for most of his children are also unknown. We do know that his first wife's name was Alice. In 1649 Alice and Moses sat in the 'third seat' at the meetinghouse, each on their respected sex's side. As seating assignments were a virtual 'pecking order' of the inhabitants, the Cox family was not a top rung family.
The couple had at least five children. John is believed to have been the oldest. He was followed by three sisters and another brother, Moses who is known to have been born on 2 November 1649. 
more about alice
By more, I actually less. If you have seen on the web in places like Find A Grave and other genealogy site, that the parents of Alice are Humphrey and Susan Wise (Wythe) don't believe it. These claims are unsourced and there is no basis for them. It is possible that Moses met Alice in Ipswich, but there is no indication of who her parents were. 
the wreck of the rivermouth
On the 20th of October 1657 tragedy struck several Hampton Families; the Swains, Philbicks and Coxes. A small ship left the Hampton harbor bound for Boston. The wreck was recorded in the Hampton records as follows:
The: 20th of the 8 mo 1657
The sad Hand of God upon Eight p[er]sons goeing in a small vessell by Sea from Hampton to boston Who wear all swallowed up i the ocian sone after they ware out of the Harbour the p[er]sons wear by name as Followeth
Sargent: Will Swaine
& Ann: Philbrick His wife
Sarah: Philbrick their daught
Alice the wife of moses Cocks:
and John Cocks their sonn:
who ware all Drowned the: 20th of the 8 mo: 1657"
Alice and John drowned in the wreck. John was probably not more than 15 years old. Why they were going to Boston is unknown. On 15 June 1658, less than a year from the death of their spouses, Moses Cox married Prudence Marston Swaine, widow of William Swaine. She was close to forty years old and had eight children. The youngest, Elizabeth, was born after the death of her father.
Moses and Prudence had one child together, a daughter named Leah, born 21 April 1661. 
John d. 1657
Mary b. abt. 1644 m. John Godfrey
Sarah m. Nicholas Norris
Rachael m. Thomas Rollins
Moses b. 2 November 1649, d. before 1682
Leah b. 21 April 1661 m. James Perkins
Moses died on 28 May 1687. If he was born in 1594 then he was ninety three years old at his death, a great age even by today's standards. I have to believe that he was not really this old, but that is just my opinion. At his death his daughters were living. His son Moses, Jr. was not named in the will, written in 1682. His wife Prudence may have been alive, but we do not know her date of death.
 Noyes, Sybil I., Charles Thornton Libby, and Walter Goodwin Davis,Genealogical Dictionary of Maine and New Hampshire, (Baltimore: Genealogical Pub., 1972) 167.
 Lechford, Thomas, approximately 1590-1644?. Note-book Kept by Thomas Lechford, Esq., Lawyer, In Boston, Massachusetts Bay, From June 27, 1638 to July 29, 1641. Cambridge: John Wilson and son, 1885.
 Hampton Towrn Records, Volume 7 page 10
 Noyes, Libby, Davis, GDMNH, 55.
Dow, Joseph, History of the Town of Hampton, N.H. from its First Settlement in 1638 to the Autumn of 1892, (Salem, Mass.: Salem Press Publ. & Print, 1893).
New Hampshire Provincial and State Papers, Vol. XXXX
 Lane Memorial Library Rootsweb page for Moses Cox
 Clifford L. Stott, "The English Ancestry of Humphrey and Susan (Pakeham) Wythe of Ipswich, Massachusetts, The American Genealogist, (Oct. 1993) 116.