Partial answer: As Christian Remling noted in remarks, it suffices to deal with the case that $(A-tI_{n})^{2} = \lambda I_{n}.$ Since $A$ is symmetric, we can only have $\lambda = 0$ when $A = tI_{n}$ for some $t$. But since the diagonal entries of $A$ are all $0,$ this only happens when $A = 0.$ Hence we may suppose that $\lambda \neq 0.$
Then $A = tI_{n} + \sqrt{\lambda}T$ for some matrix $T$ with $T^{2} = I_{n}.$
If $T = I_{n}$ we again see that $A$ is a scalar matrix, so the zero matrix.
Hence the only non-zero possibilities for $A$ arise when $T^{2} = I \neq T.$
Now trace $T$ = $r-s$ where $T$ has $r$ eigenvalues $1$ and $s$ eigenvalues $-1$
Also, since $tI_{n} + \sqrt{\lambda}T$ has all diagonal entries zero, it follows that all diagonal entries of $T$ must be equal. Hence each diagonal entry of $T$ is $\frac{r-s}{n}$ and $t + \frac{(r-s)\sqrt{\lambda}}{n} = 0.$

Now $r = s$ gives $t = 0,$ so that $A = \sqrt{\lambda}T.$ If $r \neq s,$ then $\lambda = \frac{n^{2}t^{2}}{r-s}.$

Hence the problem is reduced in essence to determining which symmetric matrices $T$ of order two have all diagonal entries equal. But if $T$ is such a matrix,
then $\frac{I+T}{2}$ and $\frac{I-T}{2}$ are mutually orthogonal idempotent symmetric matrices, each with all diagonal entries equal.

Now the problem is reduced to finding symmetric idempotent matrices $E$ with all diagonal entries equal (for we may take $T = 2E-I$ if we find such an $E$). If $E$ has rank $m,$ then note that each diagonal entry of $E$ is $\frac{m}{n}.$

We may obtain such an $E$ or each positive divisor $d$ of $n$: take $E$ to be the direct sum (in the obvious sense) of $n/d$ copies of $\frac{J_{d}}{d},$ where $J_{d}$ is the $d \times d$ matrix with all entries $1.$

At the moment, I don't see how to determine all possibilities for such an $E$.
Later edit: (...but someone else might). One observation which might turn out to be relevant is that if $E$ is such an idempotent matrix, and $X$ is a "signed permutation matrix", that is, a matrix with exactly one non-zero entry in each row and each column, that entry being $\pm 1$), then $XEX^{t}$ is another such matrix. I don't know if there are other orthogonal matrices leaving this set of idempotents invariant.

Even later edit: Note that if $E,F$ are mutually orthogonal symmetric idempotent matrices with all diagonal matrices with all entries equal, then $E+F$ is symmetric idempotent with all diagonal entries equal.

This allows us to produce symmetric idempotent matrices with all diagonal entries equal of every rank for some values of $n$. For example, when
$n = 2^{r},$ we may consider $Y = \frac{1}{\sqrt{n}} X,$ where $X$ is the character table of an elementary Abelian $2$-group of order $2^{r}.$

Let $u_{i}$ be the $i$-th column of $Y$. Then $E_{i} = u_{i}u_{i}^{t}$ is a symmetric idempotent matrix of rank $1$ with all diagonal entries $\frac{1}{n}.$ For $j \neq i,$ it is easy to see that $E_{i}E_{j} = E_{j}E_{i} = 0.$

Hence for $1 \leq k \leq n,$ we may take the sum of $k$ distinct $E_{i}$'s to get an idempotent symmetric matrix with all diagonal entries $\frac{k}{n}.$