∵y=sin4x-cos4x=(sin2x-cos2x)(sin2x+cos2x)=-cos2x,∴当x∈[-π12,π3]时,2x∈[-π6,2π3],∴-12≤cos2x≤1,即-1≤-cos2x≤12,则y的最小值为-1.故选:A.