Hi, Fannie,
The question about the New Brunswick is interesting.
Literally, it just means New Brown Village or city, Broun = associated with color, and wick + village, settlement, city.
At the time that this part of New Jersey was being settled, there had recently been a take-over of New Netherlands by the English. While the language for suggests a Dutch origin of the name, there is also, the New Brunswick, Nova Scotia, in Canada at the same time and before. So is the Brunswick a place in England, or in Scotland (Scotia), or in the Netherlans....or the one in Saxony, Germany. Hmmm!
As far as I know, the Buys families in the New Brunswick area of NJ were not "first settlers" there, nor were they in positions of authority and influence. They were single Dutch farmers, but with a definite "adventuring personality", as many of the settlers were...they would go off to find better land etc.

In my line: Jon Corneilus Buys married three times, after being widowed twice. I am from the middle wife, Phoebe Seals/Seles, dau. of an Englishman, John/Jan Seale/Seles.
Their son, Jakob Janse Buys married in the Dutch Reform Church in New Amsterdam/NYC, Marietje Joris Jakobsen Bauman/later Bowman. She was the dau. of Joris Jakobsen Bauman, farmer "at the ferry", meaning at the point where the ferry connected Long Island and Manhattan, on the L.I./Bruklyn/Brooklyn side.
It is this couple that left Manhattan/New Amsterdam/NYC to settle in the area outside of New Brunswick, NJ. At first they would have lived in New Brunswick, but farmed land outside. As the Indian threat lessened, they would have moved out to the farm to live. (An American pattern, versus the European pattern of living in a village and farming outside, but returning to the village to sleep etc.

Jakob Janse Buys and Marietje had several children. I'm descended from their son, Jakob Jakobse Buys and his wife, Neeltje Volkerts Bries. Through the first two generations in NJ, the Dutch pretty much settled down and did not move.
But, in the next generation, most of the good farm land was owned and then inherited by eldest sons, so the younger sons tended to marry and move on to new opening opportunities. Along with this came the Revolutionary War which changed and widened the opportunities for settlement elsewhere.Jakob Jakobse Buys and his wife Neeltje had several children. The two eldest sons, Jakob Jr. and Volkert Buys inherited their parents farm. They had three daughters, Wyntie, Elizabeth, and Maria. The first two remained at home, unmarried, while Maria married three times, being widowed twice. My ancestor, JOhn/Jan/Johannes Jakobse Buys was the youngest son, took whatever inheritance offered by his father, before his father's death, but after the Rev. War. and set out with his wife, Lena Marselius and their children to move to Ossabaw Island, off the harbor of Savannah, Georgia, to the south of New Jersey. At first early family researchers dismissed as myth that John Buys had actually been to Georgia, but I kept digging and found documentation that he had been a manager on a plantation on Ossabaw Island, GA after the Rev. War. While there, his wife gave birth to two children, David Fish Buys and Henry Buys (my ancestor). Lena died in childbirth (prob.) with Henry's birth. John Buys took his younger children back to NJ to his two maiden sisters to raise and returned to Georgia with his son, Edo/Eden Buys.
He later returned to the New Brunswick area and remarried to Elizabeth Gordon, dau. of Samuel Gordon and Mary Craig (Scots settlers), his sister, Maria Buys married as her third husband, Archebald Gordon, brother of Elizabeth, and Henry Buys son of John and Lena, married Margaret Gordon, youngest sister of Elizabeth Gordon and Archebald Gordon.
By that time Wyntie Buys had died, father Jakob Buys and mother Neeltje had died. The whole bunch migrated to upstate New York, leaving only Volkert Buys and Jakob Buys Jr. back on the family farm in New Jersey!
This whole migration group included the family of John Buys, his sister, Elizabeth Buys, his sister, Maria Buys Gordon, and the Gordon siblings parents, Samuel Gordon and Mary Craig. Once in New York state, they settled in the middle to the north of the state, along Lake Ontario.
In fact, John Buys' farm was right on the shore of Lake Ontario. While he and his children (a large group of children from both wives) were mainly farmers....there were suggested "other activities". There was a point of land off their farm known as "bootlegger's point"...(They seem to have been running rum, whiskey etc. back and forth to and from Canada! In John Buys house in Sodus New York, there is a "secret room" (more the size of a closet) entered only from the cellar by moving boards, that (as it was discribed to me by a local fellow) "just big enough for two barrels of whiskey!" Much later, after the Buys family no longer owned this house, it was used to hide run-away slaves as a "depot" on the famous "Underground Railroad" to get fugitive slaves safely into Canada!
Ok, why all this LONG story. It is just that the Buys family were know to be very adventurous, to trade in rum and whiskey, etc. As I look at the whole picture, I have no proof, but it seems that John Buys did not go to Ossabaw so much with a strong desire to managed a plantation for a rich Ossabaw family, as it was an opportunity to have access to the running of rum in from the Caribbean etc.
So it is not as wild as it may seem that at some point one of the members of this family went off the Caribbean and then on to SA! If this was the case of Berand Buys, he would have "dropped off the map" as far as a paper trail back home as those who stayed behind left, and thus be missed in the history of the Buys family research here.
With John Buys' death, several of the sons and dau. migrated further west as quiet farmers, leaving the more adventurous style behind. My line with Henry Buys and Margaret Gordon and Henry's brother, Daniel Fish Buys both moved to Erie, Co. Pennsylvania as that area opened, while sons of John Buys and Elizabeth Gordon, migrated to Michigan as that state opened.
The OTHER Buys family of the New Brunswick are of NJ, that of the descendants of John/Jan Buys, son of Jan Thyssen Buys, (who we know had Caribbean connections, moved further west in NJ and then after the Rev. War, on to Tennessee, Kentucky, and further west.
I know that this is more than you ever wanted to know about our Buys families, but we may still eventually find connections.
Elsie