The major argument for this is that people should not be encouraged to come to the US illegally. Providing them with health coverage will encourage them and therefore we should not do it.
Coming to the US illegally is, by definition, against the law. In any democratic society, we should encourage people to obey the law, not to break it. If we create benefits that will be given to people who break the law, we are encouraging them to act illegally. If we do this, we are acting unethically because we are encouraging people to ignore the law. We are also acting in an impractical way because we are encouraging more illegal immigrants to come at a time when we cannot really afford to have them here.