Is it illegal for non-native English speakers to work at hagwons?
I don't think it's illegal, hakwons just won't hire non native English speakers. They want teachers from certain countries.
It is not illegal to 'work' there but you must have the appropriate visa to be in the country and work.
You are not allowed to work at jobs that don't correspond to your Visa. For example, to be a 원어민 "Native Speaking Teacher" you must have an E2 Visa. Of course F-visas are ok too but both must have approval from the Busan Office of Education.
Not all schools report to the BOE and that is illegal if they are advertising a Native Speaker. Some schools, like one in Centum City I cannot name because I don't want to get sued, lies to parents about where the teacher is from but still registers them. For example, I know that when they have a South African teacher (they've had a few), they tell parents they're from the UK because they think parents don't want someone from Africa teaching their kid. However, South Africa is one of the designated countries that can get an E2 visa.
I also know a few Filipinos and Koreas who were told to lie about where they were from.
USA, Canada, UK , Australia, New Zealand, Ireland and South Africa only.