= Solution
Multiplication gives
$$
B^TB=(I+nm^T)(I+mn^T)=I+mn^T+nm^T+nn^T.
$$
The vector $p=m\times n$ is an eigenvector with eigenvalue one. On the plane spanned by the orthonormal pair $m,n$, the matrix is
$$
\begin{pmatrix}1&1\\1&2\end{pmatrix}.
$$
Its eigenvalues are the roots of $\mu^2-3\mu+1$, namely
$$
\mu_\pm=\frac{3\pm\sqrt5}{2}.
$$
For either root, an eigenvector is $m+(\mu_\pm-1)n$. Therefore
$$
\boxed{
\begin{array}{c|c}
\text{eigenvector}&\text{eigenvalue}\\ \hline
m\times n&1\\
m+\frac{1+\sqrt5}{2}n&\frac{3+\sqrt5}{2}\\
m+\frac{1-\sqrt5}{2}n&\frac{3-\sqrt5}{2}
\end{array}}
$$
These positive eigenvalues are the squared <singular values> of $B$.
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