Proving $|1 + z_1|^2 + |1 + z_2|^2 + \ldots + |1 + z_n|^2 = 2n$

The equality has a simple geometric interpretation. If $\,G\,$ is the centroid of $\,n\,$ points $\,P_k\,$ it is a known property that $\,\sum WP_k^2 = n \cdot WG^2 + \sum GP_k^2\,$ for any point $\,W\,$ $\left(\dagger\right)\,$.

Taking $\,P_k\,$ to be the points with affixes $\,z_k\,$ in the complex plane, the condition $\,\sum z_k = 0\,$ means that $\,G \equiv O\,$, and therefore $\,GP_k = |z_k| = 1\,$. Writing the previous equality for a point $\,W\,$ of arbitrary affix $\,\omega \in \mathbb C\,$ reduces to $\,\sum |\omega-z_k|^2\,$ $\,= n \cdot |\omega|^2 + \sum 1 = n\left(|\omega|^2+1\right)\,$, and the equality in OP's question follows for $\,\omega = -1\,$.

[ EDIT ] $\;$ To answer the solution-verification part of the question, the posted proof is correct. In fact, it would be straightforward to adapt it as to prove the more general result derived here.

I assume the "taking the modulus of both sides, we obtain ..." line is a transcription error. Given what follows, it was obviously supposed to be "taking the conjugate ...", instead.

[ EDIT #2 ] $\;$ Prompted by boojum's comment, these are several other contexts where the same relation $\,\left(\dagger\right)\,$ occurs in some form.

• in geometry, proving that the locus of points in the plane with constant sum of squared distances to a set of fixed points is a circle  e.g. [1];

• in statistics, proving that the mean minimizes the squared error  e.g. [2];

• in mechanics, the parallel axis theorem about moments of inertia.

I would write $e=(1,....,)^T, z=(z_1,...,z_n)^T$ and then note that $e^T z = 0$ and $\|e+z\|^2 = e^T e + e^T z + z^*e + z^*z = e^T + z^*z = 2n$.