He can make it illegal, but he can never completely shut down homeschooling. As long as I have breath in my body, I will teach my own children. They attend public schools, but their education is far more comprehensive than what they are taught there. Their understanding of history and politics is greatly affected by the experiences of their family members and by our dinner table conversations over the years. Nobody can take that away unless they physically remove our children from our custody.
My parents fled Czechoslovakia after the communist takeover, while my grandparents, aunts and uncles stayed behind. I saw how they lived, and how my cousins were educated. In times of oppression, we may need to be more creative in how we pass along our values to our children. The morning after the election, I woke up thinking about an onion. It grows underground, out of sight, and needs little protective shell as it is protected by the earth. Similarly, some aspects of our lives may become like that onion - out of sight but quietly growing, until the time is right to emerge fully formed.
He can't. We won our victories, up to and including a Supreme Court ruling. Children are not wards of the state, and parents have the right to direct their education in any way they see fit.
Homeschooling is, by and large, a state matter. However, we need to stay ever, ever vigilant that our state legislatures will not pass stupid laws encroaching on our freedoms.