Bounds on the number of squares in recurrence sequences: $y_0=b^2$ (I)

We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, ${\left(y_k\right)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}$, arising from the solutions of generalised negative Pell equations, $X^2-d{Y}^2=c$, where $-c$ and $y_0$ are any positive squares. We show that there are at most $2$ distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most $5$ distinct squares when $y_0=b^2$ for infinitely many values of $b$, including all $1\le b\le 24$, as well as once $d$ exceeds an explicit lower bound, without any conditions on the size of such squares.