Step 1: Understanding the Concept:
The hybridization and geometry of coordination complexes depend on the oxidation state of the central metal and the strength of the surrounding ligands (Crystal Field Theory).
Step 2: Detailed Explanation:
1. Oxidation State: In $[Ni(CN)_4]^{2-}$, Cyanide (CN⁻) is -1. So, $x + 4(-1) = -2 \Rightarrow x = +2$. Ni is in the +2 state.
2. Electronic Configuration: $Ni^{2+}$ is $3d^8 4s^0$.
3. Ligand Strength: CN⁻ is a strong field ligand. It forces the electrons in the $3d$ subshell to pair up.
4. Result: After pairing the 8 electrons in $3d$, one $3d$ orbital becomes vacant. This vacant $d$ orbital, along with one $4s$ and two $4p$ orbitals, hybridizes to form four dsp² hybrid orbitals.
5. This leads to a square planar geometry and a diamagnetic complex.
Step 3: Final Answer:
The hybridization of Ni in $[Ni(CN)_4]^{2-}$ is dsp².