Cizikas, 23, got a two-year contract. He scored six goals and 10 assists last season, his second full year with the club. In 2013, he had a game-winning goal in a playoff series against Pittsburgh.
The 23-year-old de Haan received a three-year deal. He scored three goals and had 13 assists in 51 games as a rookie, and showed solid improvement throughout the season.
He began the 2013-14 season with Bridgeport of the AHL before earning a call-up to the Islanders on Nov. 28.
The Islanders selected de Haan in the first round, 12th overall, of the 2009 NHL draft.