NHI pays for 70% of your expenses on whatever it covers. Obviously, it doesn't cover all possible medical situations.
Your first year, NHI costs only 2500 yen/month because the government has no record of your earnings here. After that, NHI costs go up about tenfold. Live with it. Everyone else does. The actual cost depends on last year's Japanese earnings, your location, and a few other factors. Getting out is a little hard, but just get another form of insurance, and stop paying. The only risk you run is that if you want to go BACK to NHI, you are obligated to make backpayments up to 2 years worth.
Go with whomever you want, but here are 2 caveats.
1. If your company has corporate insurance (a different form of NHI called shakai hoken), you are probably obligated to take it. It is more than just insurance; it is also a retirement and pension plan, but don't get upset if you plan NOT to retire here. You can get back most of about 3 years of those payments.
2. If you can go with another insurance outfit, beware that they may ask that you pay all of the initial costs of any treatment and medication up front. 100%. Then, they will reimburse you, but that could take a month or more.