Le Maohua, “On Mersenne Numbers” [in Chinese], *Journal of Jishou University (Natural Science Edition)* 20(1) (March 1999): 17–19, shows that the squarefree part of $2^p - 1$, with $p \ge 11$ prime, is greater than $(\pi{}p/log\thinspace{}p)^2$.

(Added in 2023:) Since Le’s paper is not widely available, I will attempt some explanation of it, working from notes in English that were provided to me privately and which I am not in a position to share. Le was apparently the first to notice the relevance to this problem of a paper of Wilhelm Ljunggren, “Über die Gleichungen $1 + Dx^2 = 2y^n$ und $1 + Dx^2 = 4y^n$,” *Det Kongelige Norske Videnskabers Selskab Forhandlinger* 15(30) (1942): 115–118, on the Diophantine equation $1 + Dx^2 = 4y^n$. The case $D = a$, $x = b$, $y = 2$, and $n - 2 = p$ (with $p$ prime) corresponds to a Mersenne number factorization in the form proposed in the original post above, $M_p := 2^p - 1 = a \cdot b^2$.

Now if $h(\cdot)$ represents the imaginary quadratic number field $\mathbb{Q}\left(\sqrt{\cdot}\right)$, then by Gauss&rsquo;s theory of classes, $h_p(-M_p) = h(-b^2a) = h(-a)$. After demonstrating the impossibility of a square divisor of $M_p$ for the small cases $11 \le p < 101$ by refering to published factorizations of the Mersenne numbers, Le uses Ljunggren's result that $p - 2$ divides the class number $h(-a)$, and estimates of the class number by Louboutin and others, to establish a lower bound on the squarefree part of $M_p$ in the remaining cases.