No, $n$ is not squarefree. This is a 1937 result of Steuerwald:
R. Steuerwald, "Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl," S.-B. Math.-Nat. Abt. Bayer. Akad. Wiss., 1937, pp. 68-73.
A very nice recent paper on this theme is by Fletcher, Nielsen, and Ochem (Math. Comp.); the preprint version is freely available on Nielsen's website: https://math.byu.edu/~pace/OPNSieves_web.pdf