I know that when a place of employment sponsor you for a job here they are technically responsible for you. But are they required by law to provide you with some type of health insurance while you work for them? I am especially interested in the E2 visa and whether the school must provide with health insurance.
Yes, your school or sponsor is responsible for providing you with health insurance. Every contract should have a clause mentioning it, so I really wouldn't be too concerned. Typically, you'll be paying half of the monthly payment which works out to between 50,000 to 70,000 won depending on your situation. Stop by the NHIC site to get a little background on the program. It's pretty comprehensive, so I really don't need to go into too much detail other than saying it's good, it's cheap and you'll miss it once you leave. I happen to think that the Korean system is one of the finest in the world. True, it might be going bankrupt, but it has been excellent for me.