If I weren’t Japanese, I might .........?

Yeah apparently both are correct. Guess what?!?! I'm American.

If I weren’t Japanese, I might have needed a visa to enter the country.

If I weren’t Japanese, I might need a visa to enter the country.

Both of these sentences are correct, but the meanings are a little different.

"If I weren’t Japanese, I might have needed a visa to enter the country." This seems to say that I already went to Japan or that I have been to Japan or that I am in Japan now, and because I'm Japanese I didn't need a visa to get into the country.

"If I weren’t Japanese, I might need a visa to enter the country." This implies that I'm planning a visit to Japan, and because I'm Japanese I won't need a visa to get into the country.

The restructured sentence is given below .

If I weren't a Japanese, I might have needed a visa to enter the country.

First

Technically yes, however the word "weren't" refers to more than one person so I would change it to "wasn't" or better still to it's full glory of "was not" to be grammatically correct.

both are correct