Earlier quoted context omitted.
From the very specificity of the restriction, I gather that they're not copyleft-adverse. They're "Don't upload stuff that risks us being forced to divulge our closed-source code.". Without the prohibition, even if a user of the service included some AGPL libraries without Square's knowledge, a cunning lawyer could argue that Square was complicit in distributing this code to end-users and thus liable. The prohibition…
A cunning lawyer could argue anything. But, when did Square ever agree to the copyright license? Who did the actual copyright infringement? And AGPL isn't the only "must publish all your server's source code" license available, so to prohibition is too specific if that's the actual reason.
This is why we have the DMCA safe harbor provisions - it'd be incredibly easy to be on the hook for copyright infringement as a company hosting any user content otherwise. Those safe harbor provisions aren't a blank check to knowingly commit wanton copyright infringement however, as can be seen with all the legal wrangling over Megaupload, as an example [1].
Square knows about the AGPL, knows they might not be abiding by it's terms if they redistribute AGPL code, and knows they probably don't have the right to redistribute under any other term. They can choose to knowingly possibly violate the license, or to knowingly commit copyright infringement, or to ban AGPL code. Those first two options, with that "knowingly" in there, open them up to potential legal action.
> And AGPL isn't the only "must publish all your server's source code" license available, so to prohibition is too specific if that's the actual reason.
If you make Square's lawyers aware of some of those other licenses, it wouldn't suprise me if they get banned too for the same reason. It's not like they have psychic knowledge of the existence of all licenses. Failure to audit the universe doesn't mean that's not their reason.
[1] https://en.wikipedia.org/wiki/Megaupload#Safe_harbor_provisi...