If you can keep calm while they're reacting strongly, try gently asking them to sleep on it for 24-48 hours... ask them to give themselves a little bit of time to come to grips with what they've been confronted with. If after that time, they really still are upset that the reading went there, I'd return their payment (and also note the problem so that if they come back again, we work out up front that payment is for time spent and if they're not ready to face what comes up in a reading, it's ok not to seek one out).
Most of the time, initial reactions are about shock, fear, embarrassment at being exposed... there's an initial strong rejection of something they weren't prepared to deal with right then, but with a little bit of time thinking it over, it starts things cooking in their head and lets them start to get a handle on it. Which, I think, is exactly why it came up in the reading in the first place.
As a reader, I think we have to be prepared to be the surrogate target for a lot of deep chaotic feelings that surface because of the reading. All you can do is try (hard hard hard) to keep the focus on the seeker and the reading and try to regard that high emotion as being about the subject of the reading and things happening inside them, and not about you even if that's how they word it.
Sometimes if you give them a little time to process what's happened, that high emotion turns out to be a breakthrough moment where they can start to make positive changes in their life. It's not fun, it can be draining, but I'd rather someone getting incensed than the ones that show nothing at all and the reading seems to drop like a thud onto the ground. (and those also sometimes surprise you when later on they say it really helped and how... but during, those are painfully difficult for me)