The Christian may take up any lawful profession and teaching is certainly a lawful profession. However, the Christian may never discharge the duties of their profession in a dishonest or sinful way. For example, a Christian may be a fine pharmacist and compound excellent medicine. However, if the order came down from "higher up" for the pharmacist to incorporate small amounts of toxic or poisonous substances into their compounds for no good reason, I hope it's plain why the Christian pharmacist must refuse.
I think teaching in the public schools would present a very similar problem. There may be many public schools or classrooms where Christian teachers can teach good things and keep error out of their curriculum. However, isn't it obvious that teaching error will be more and more required of public school teachers in the coming days? Isn't it obvious that public school teachers will soon be required to call boys girls and girls boys? Won't they be subject to penalties if they counsel a girl against killing her unborn child? A school teacher is an authority figure and thus accountable to God for the ones under their instruction. Can the Christian public school teacher stand by and allow what they know to be poison incorporated into what they teach? I don't know, maybe there are still school districts where a Christian teacher can teach creationism as fact and evolution as fantasy, but I suspect not.
Yes, this about sums up my thoughts exactly.
And it is my understanding that, while the government continues to hand out "free" education, private schools will be unable to pay their teachers comparable salaries. Of course, there are the exceptions, but it isn't entirely encouraging for someone who desires to teach middle/high school ages as a vocation.