Cizikas, 23, registered six goals and 10 assists for 16 points in 80 games this past season with the Islanders.
In 140 career NHL games, the Toronto native has accumulated 12 goals and 13 assists for 35 points.
The Islanders selected Cizikas in the fourth round (92nd overall) of the 2009 NHL Entry Draft.
de Haan, 23, picked up three goals and 13 assists for 16 points in 51 games this past season as a rookie with the Islanders.
The Islanders selected de Haan in the first round, 12th overall, of the 2009 NHL Draft.