Is there a way to find old passport records via the government?

You can make a request for entry/exit records (?????) from the Japanese Ministry of Justice, but for foreigners records are only available from November 1st 1970 onwards. The process looks bureaucratic in the extreme (details here, Japanese only) and you'll almost certainly need to engage someone in Japan to help you, since the application forms are all Japanese and you need things like Japanese tax stamps to pay for it. You will also have the additional complication of establishing that you're a legal representative (?????) authorized to access their records, which will be tough since the documentation for this seems to assume that you're a Japanese citizen with a koseki that can be used to demonstrate the relationship.

For China, I have no idea, but given that the 1950s and 1960s were the era of the upheaval of the Great Leap Forward, the Cultural Revolution etc, I would imagine you'd have extreme difficulty locating any records even if they survived. In fact, since the PRC was pretty much closed to the world at this time, are you sure they actually visited Communist China, or did they go to eg. Hong Kong or Singapore instead?

If you know whether your relative traveled by ship or plane and know where they departed from, and assuming they left from the US, a more fruitful resource might be the INS's archive of Passenger Arrival Records. However, you'll need to have a pretty good idea of where exactly to look in those piles of microfilm, that site doesn't give the impression there's a searchable digital archive or anything.