Hello Abairs and Abares. The ancestors you seek prior to coming to the US are HEBERTs, a venerable name in French Acadian history. You are descended from one of the earliest pioneers of New France. Early 1600s. I believe they were found early on in Acadia (now Nova Scotia, more or less) and in early Quebec. In 1755 and succeeding years the English were bent on an ethnic cleansing of the Acadian people, and deported them to all the English colonies, English jails, Australia and other places, breaking up families, and in general causing untold grief, loss, despair and deaths.Nonetheless, many survived, and some went to Quebec, after being released from exile, some in 1768. After going to Quebec, New Brunswick, etc., they had their marriages and baptisms, which occurred in Protestant exile, "rehabilitated" in Canada, so that you get peculair dates, such as children born before parents are married in the Catholic church, etc.
You might try contacting the American Canadian Genealogy Society in Manchester, NH (They have a small online site). For $25 a year you can be a member, recieve a journal, post queries, etc. And for $20 a line, volunteer genealogists will trace your family tree back to the 1600s, money back if no success. A deal you can't beat.Get on a search engine and type in Acadian Genealogy or Acadian History. You will soon learn you are descended from real survivors, as I did. Good Luck. And see if there is a Hebert forum. Hebert in French sounds like Abair, by the way. There are some Abares in Ashland, NH, where some of my mother's family lives.