Vincent would have been considered an automatic selection as an opener after making scores of 52, 224, 13 and 92 in his last four Test innings, as well as a superb century in Wednesday's One-day international against the touring West Indies.
But coach John Bracewell said Vincent was overlooked after he told the selectors he wanted to bat in the middle-order but they could not find a place for him.
''Lou continually said that he didn't see himself as an opener and would prefer to bat in the middle order,'' Bracewell told New Zealand Press Association.
''We weren't prepared to play guys who were reluctant.'' Vincent's omission leaves New Zealand facing the prospect of fielding a new opening partnership against the West Indies with the uncapped Jamie How the only chosen specialist opener.
Hamish Marshall, who normally bats in the middle-order, is likely to be promoted up the order from three with Peter Fulton, who is also uncapped, named in the middle order.
A third uncapped player, offspinner Jeetan Patel, was also named in the squad while pace bowler Chris Martin was recalled.
The first Test starts at Eden Park on Thursday, with the second and third Tests to be played at Wellington and Napier.
New Zealand squad: Stephen Fleming (captain), Jamie How, Hamish Marshall, Peter Fulton, Nathan Astle, Scott Styris, Brendon McCullum, James Franklin, Kyle Mills, Daniel Vettori, Jeetan Patel, Shane Bond, Chris Martin.