I need to elaborate - I missed that the law requires private schools as well, so I may be in agreement with you. If the school is funded by my tax dollars, then I have no problem with being required to pledge allegiance to our nation (which gives them the money in the first place)! Otherwise, they can home school or go private.
A private school, however, should not be required to do this at all, but I would hope they did anyway.
Education is mandatory. People have no choice but to go, at gunpoint if necessary. In government schools under this law, so is pleadging allegience out loud, even if you don't feel allegience. Therefore you are being forced at gunpoint to pledge alligence.
Then they try to force people in private schools to do the same. Neither is right. And when I educate my own children, I still have to pay for yours.
Substitute another country, like Iraq or Nazi Germany into the equation and remove this country and see if you think it's a good idea then.
If Hillary were to pull a sucessful coop and write one with her name in there, see if you agree then.
Government schools are a prominent part of the Communist Manifesto. They are incompatible with a free society in the long term.