Statement (I): This statement is incorrect. Dimethyl glyoxime forms a square planar complex with nickel, not a six-membered covalent chelate, when treated with NiCl$_2$ solution in the presence of NH$_4$OH. The resulting complex, known as a nickel-dimethylglyoxime complex, is characterized by a square planar structure where the nickel is coordinated by the nitrogen and oxygen atoms of two glyoxime molecules.
Statement (II): This statement is true. Prussian blue (Fe$_4$[Fe(CN)$_6$]$_3$) is a coordination compound containing iron in both +2 and +3 oxidation states. The compound forms due to the combination of Fe$^{3+}$ ions and [Fe(CN)$_6$]$^{4-}$ complex ions, leading to a structure with mixed oxidation states of iron.
Conclusion: Statement I is false, and Statement II is true.