Maybe.
The OP doesn't say if they will be working for a Japanese or US company whilst in the UK, or for a UK company. They also don't mention whether the intended final system for retirement will be from Japan or the US (US SSA). They don't mention if they have been posted to the UK for a US company for less than 5 years, and if so, will the US company sign a certificate of coverage. If it's a Japanese company (and the Japanese retirement system), there appears to be a UK/Japan agreement; which is why I included the HMRC link. I'm assuming the OP name of "jrb in japan" means they're working there now. Of course, as always, I could be wrong.
If the question is simply since the UK company they'll be employed by is paying a private health insurance benefit and pension, do they have to pay NI in addition, then I agree, the answer is: yes, they must pay NI.
More info required, but the results for all the above are available on the linked sites.