Possible eigenvalues of matrix $C = A(B^TA)^{-1}B^T$

$B^T x=0$ is possible for nonzero $x$. A simple example comes from taking $$ A=B=\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}. $$

This meets all the desired properties and the matrix $C$ has zero as an eigenvalue.

Also note that if $A$ and $B$ are $p \times n$ and $ B^T A$ is invertible, then we must have $n \leq p$, since $A$, hence $B^T A$, has rank at most $\min (p,n)$ and $B^T A$ is $n \times n$. Furthermore, the rank of $C$ is at most $n$, since the rank of $A$ is at most $n$. It follows that $C$ always zero as an eigenvalue with geometric multiplicity at least $p-n$.