High-index dielectric resonators support different types of resonant modes. However, it is challenging to achieve a high-Q factor in a single dielectric nanocavity due to the non-Hermitian property of the open system. We present a universal approach of finding out a series of high-Q resonant modes in a single nonspherical dielectric cavity with a rectangular cross section by exploring the quasi bound-state-in-the-continuum (QBIC). Unlike conventional methods relying on heavy brutal force computations (i.e., frequency scanning by the finite difference time domain method), our approach is built upon Mie mode engineering, through which many high-Q modes can be easily achieved by constructing avoid-crossing (or crossing) of the eigenvalue for pair-leaky modes. The calculated Q-factor of mode TE(5,7) can be up to Q_theory = 2.3 × 10⁴ for a freestanding square nanowire (NW) (n = 4), which is 64 times larger than the highest Q-factor (Q_theory ≈ 360) reported so far in a single Si disk. Such high-Q modes can be attributed to suppressed radiation in the corresponding eigenchannels and simultaneously quenched electric (magnetic) field at momentum space. As a proof of concept, we experimentally demonstrate the emergence of the high-Q resonant modes [Q ≈ 211 for mode TE(3,4), Q ≈ 380 for mode TE(3,5), and Q ≈ 294 for mode TM(3,5)] in the scattering spectrum of a single silicon NW.