How a unitoid matrix loses its unitoidness?

A unitoid is a square matrix that can be brought to diagonal form by a congruence transformation. Among different diagonal forms of a unitoid $A$, there is only one, up to the order adopted for the principal diagonal, whose nonzero diagonal entries all have the modulus $1$. It is called the congruence canonical form of $A$, while the arguments of the nonzero diagonal entries are called the canonical angles of $A$. If $A$ is nonsingular then its canonical angles are closely related to the arguments of the eigenvalues of the matrix $A^{-*}A$, called the cosquare of $A$.
Although the definition of a unitoid reminds the notion of a diagonalizable matrix in the similarity theory, the analogy between these two matrix classes is misleading. We show that the Jordan block $J_n(1)$, which is regarded as an antipode of diagonalizability in the similarity theory, is a unitoid. Moreover, its cosquare $C_n(1)$ has n distinct unimodular eigenvalues. Then we immerse $J_n(1)$ in the family of the Jordan blocks $J_n(\lambda)$, where $\lambda$ is varying in the range $(0, 2]$. At some point to the left of $1$, $J_n(\lambda)$ is not a unitoid any longer. We discuss this moment in detail in order to comprehend how it can happen. Similar moments with even smaller $\lambda$ are discussed, and certain remarkable facts about the eigenvalues of cosquares and their condition numbers are pointed out.