Most teaching jobs are, of course, with government schools. Most government agencies are not allowed to sponsor people for visas, and probably couldn't afford it if they could. The one exception I see over and over are for medical personnel, especially nurses. Unfortunately they only recruit nurses that are qualified under the American system (mainly from the Philippines and Panama) and therefore have already passed their boards.

However, there are no restrictions on teachers in private schools. These jobs tend to pay less, since the teachers should have an easier time of it with private students (better behaved, study, probably come to school unarmed).