This is a French name, either was "du chêne" (from the oak) or less likely "du chaîne (from the mountain chain). The name could be literally "from the place where the oak tree[s] grew" or "from the mountains", or it could have developed a figurative meaning like "solid/strong/unyielding" "like an oak" or perhaps "tall/towering/formidable" "like a chain of mountains".
The 'e circumflex' (ê) in French is used when the letter 's' that used to come after the 'e' became silent and was dropped (so original spelling was "du chesne" = sheZ-nuh, but changed over time into "sheh-nuh").
The modern English forms Dushane, with the emphasis on the last syallable (like the original French) or Dushen (where the 'du' is emphasized) and other variants like DUCHIN (the society pianist and bandleader Eddie was one) are all really from the same source.
In French, the plural "deschênes" (day-SHEN) is still a very common surname.
As others have already said here, many people with these surnames are French Protestant, or Huguenot (oo-ghen-oh) in origin. Huguenots came from the French provinces of Alsace (al-SASS) or Lorraine along the German border and were forcd to flee from France at least three times between the 1500s and late 1700s when French kings became nervous about their growing numbers and increasing influence. Most went first to the German states (there was no Germany until 1870), Holland (which included modern Belgium then), or to England, Wales, or Ireland. From these points many later made their way to the American Colonies before the Rev War. The English were happy to have them settle Ireland to counteract the large number of Catholics there; later they encouraged them to settle northern USA and Canada for the same reason.
You can find lots of info about Huguenots all over the world by using any search engine. Another interesting source is right here on GENFORUM. Just go to the listing for each country in the world, select the ones from Western Europe, and put HUGUENOT in the upper left block on each front page, then click GO. All the listings mentioning them will pop up (main problem with this word is that most people can't spell it properly), so you might miss some of the postings).