I had not heard of a supermajority requirement for certain state supreme court decisions before I came across this article:
Nebraska has a supermajority clause in its constitution.
The requirement that five judges hold a law unconstitutional in order to strike it down is found in NEB. CONST. art. V § 2:
The supreme court shall consist of seven judges, one of whom shall be the Chief Justice. A majority of the judges shall be necessary to constitute a quorum. A majority of the members sitting shall have authority to pronounce a decision except in cases involving the constitutionality of an act of the Legislature. No legislative act shall be held unconstitutional except by the concurrence of five judges.
See above article.