I'm not talking about fraud here, or things like that. I'm talking about when you get into a situation which can only cause the company harm and embarrassment, the generally appropriate thing to do is remove your link to the company.
If he was a random employee, sure, they don't pay you to do that, but executives, yeah, this is one of the reasons they are paid well.
I'm sure he also holds enough options/stock that he has a financial incentive to not cause the company financial trouble through his alleged personal indiscretions.
If he was sued personally, but Square was not, then yes, cutting this link probably would help.
This is consistent with the Square statement: "While we have not found evidence to support any claims, Keith exercised poor judgment that ultimately undermined his ability to remain an effective leader at Square."
In other words, both Square and Rabois suggest that what transpired was an error in judgement which would complicate things if he continued at Square, but not legally actionable. Since both Rabois and Square can continue independently, minimizing the complications with a resignation can make sense even if completely 'innocent' of legally-enforceable damages.
I'm guessing they offered him the choice of resign and possibly vest a few more options or not resign and be fired by square.