Reading the complaint, ADP is not likely come close to winning. The defamation they claimed happened is that Zenefits "alleged that ADP intentionally sought to cause harm to ADP’s clients solely to gain an unfair competitive advantage against Zenefits.” Even if you argue this is defamation per se (and not per quod), about a matter of private concern, they'd still have to prove "4. That [name of defendant] failed to u…
Basically, Zenefits accuses ADP of blocking access solely because of a single traffic spike. On top of that, it accuses ADP of changing their reasons for cutting off Zenefit's access. They then accuse ADP of "spreading FUD"--and lying--about Zenefits for the purposes of quashing the competition.
That last statement, especially, is pretty dangerous under defamation law as it's pretty close per se defamation, in which case ADP would only have to prove the statement was made in court to win. There's no defense to that except proving that ADP was lying about Zenefits.
Based on what both parties have said, and what we know about how Zenefit's system actually works, a jury could definitely find that Zenefits defamed ADP and I would say, based on prior experience with juries, that there is at worst a 50/50 chance from ADP's perspective if this goes to trial.