Various variants of the matrix
$$A_{J,K} = |J\cap K|$$
were studied, and the spectrum was computed. A one-parameter variant is given by
$$A^{(i)}_{J,K} = \binom{|J\cap K|}{i}$$
for some fixed $i$, and your problem corresponds to $i=1$. In total, six variants are given in Section 3 of [this friendly paper][1] by Ghareghani, Ghorbani and Mohammad‐Noori.  In particular, the spectrum of $A^{(i)}$ is described in Lemma 9. The lemma is attributed to [R. M. Wilson (1982)][2], who uses somewhat different notation and terminology. The case $i=1$ recovers your theorem, and in general the non-zero eigenvalues are given by 
$$\lambda_j=\binom{s-j}{i-j}\binom{n-i-j}{s-i}, \qquad j=0,1,2,\ldots,i,$$
with $\lambda_j$ having multiplicity $\binom{n}{j}-\binom{n}{j-1}$ (note that $\binom{n}{-1}:=0$). The eigenvalue 0 has multiplicity equal to the remaining dimension, i.e. $N-\binom{n}{i}$.

Other useful references, including to additional works of R. M. Wilson, appear in the bibliography of the G-G-MN paper.


  [1]: https://onlinelibrary.wiley.com/doi/full/10.1002/jcd.21308
  [2]: https://www.sciencedirect.com/science/article/pii/0024379582900271