That's what I thought. Having the partner present seems to be the right way of handling this. The company is responsible for employees well-being and shouldn't let a client bullying them.
We weren't in the meeting, but if the person cried, there's a possibility that the person was actually bullied. If the work sucks, it can be discussed outside of the meeting with the person's manager for instance. It's not because the person works for McKinsey and is an inexperienced ivy league graduate that it's ok to be mean.
Why would they fire him after a singe incident?
Sounds like McKinsey is a more companionate organization than you, and that's saying something:)