I can't imagine that this law would be terribly effective. If I were a parent in New Jersey and I were home-schooling my children, I would simply tell the authorities that my kids were living with a relative in another state.
And when the New Jersey authorities contact my relatives to ask for verification of their school attendance, they would simply reply: "We're home-schooling them. You don't have any jurisdiction here anyway, so get the hell out of my face."
And when the New Jersey authorities contact my relatives to ask for verification of their school attendance, they would simply reply: "We're home-schooling them. You don't have any jurisdiction here anyway, so get the hell out of my face." If these children are deductions on IRS 1040 forms they will be expected to be at the address where the deduction is taken.
Big Brother is everywhere! The system knows all.