As to their health insurance, if it is a Japanese insurer they should really check if the insurer is set up to handle an American claim, or if they have a US branch or a collaborative agreement with an American health-insurer to receive direct billing from the hospital/doctors: traditionally many US health providers have trouble billing to out-of-state insurers/HMOs, never mind out-of-country. In this case they should be able, once they’re settled in a bit, to shop for providers willing to make arrangements ahead of time: he should check with the personnel/employee benefits department at the company, who probably have expertise in dealing with this and maybe even a list of providers who are willing to work with the foreign insurer. He should be prepared to pay some large bills up-front (specially such things as lab tests, specialist referrals and prescriptions, that happen outside the primary care provider/hospital) then request reimbursement.
As for citizenship, like smiling bandit said, the US is a jus solis jurisdiction so unless they were under diplomatic immunity, Baby will be born American. And since both parents are Japanese citizens, s/he will almost certainly be one as well, by jus sanguinis – meaning no problem carrying on life as a regular Japanese citizen back home if they so choose. Don’t know Japan’s position on “dual citizens”’, the USA does not encourage it or technically “recognize” it, but more like “looks the other way” until and unless an actual problem arises.