In general, those enrolled in the National Health Insurance system are those employees not covered under the health plan of the company that they work for, people who are self-employed, and the unemployed. These people are signed up in case of an emergency illness. As this insurance is for residents of cities, towns and villages, non-Japanese are basically eligible for it the same as Japanese citizens. However, not all foreigners can enroll in the system. For example, because there is no set medical treatment system in your country, if you were to undergo medical treatment in Japan on a tourist visa having just paid a small fee to be enrolled in insurance scheme and as the system is intended for Japanese citizens, the possibility exists that the system could collapse. That having been said, let's take a look at the situations in which a non-Japanese can be enrolled in the National Health Insurance system.

You are a registered non-Japanese and received a period of residence of more than a year when you entered the country.

However, even if your initial period of residence is less than a year, if you can clearly prove that your reasons for entering Japan were to stay for more than one year, you'll likely be able to enroll. In this case, according to your residence qualification, you will need to submit various material and prove that you will be staying more than one year.

I'm not familiar with all the details of your situation, but seeing as you have lived in Japan for three years, it would be a good idea to head down to your local municipal office and ask for more information on getting enrolled.