The eigenstates of the Nth powers (N≥3) of the annihilation operator of the non-harmonic oscillator are constructed, and their mathematical and higher-order squeezing properties are studied. The results show that they form a complete Hilbert space, and the Mth power ［M=(n+1/2)N; n=0, 1, 2, …］ squeezing effects exist in all eigenstates when N is even.