In the quadratic sieve algorithm, the idea is to find $a$ and $a$ such that $a^2 \equiv b^2 \bmod n$. We need that $a\not\equiv \pm b \bmod n$. However, there the $c$ is not necessarily $1$. $\gcd(b \pm c,n)$ returns non-trivial factors.
However, in Shor's algorithm, we specifically need to find square roots of $1$ (in modulo $n$) i.e. we look for $a$ such that $a^2 \equiv 1 \bmod n$. That is, $b$ is specifically $1$. Why is this choice necessary?