Putnam 1991 B5

Let $p$ be an odd prime and let ${\mathbb{Z}}_p$ denote (the field of) integers modulo $p$. How many elements are in the set $$\{x^2: x \in {\mathbb{Z}}_p\} \cap \{y^2 + 1 : y \in {\mathbb{Z}}_p\}?$$