Sharp bounds on the number of squares in recurrence sequences and solutions of $X^2-(a^2+b)Y^4=-b$

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^m$, $2p^m$ or $4p^m$, where $m$ is a non-negative integer, $p$ is prime, $\gcd (a^2,b)$ is squarefree and $X^2-(a^2+b)Y^2=-4$ has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations.