They could, but the court would be interested in the school officials' motives. Back in 1995, the justices struck down an Alabama law that permitted a moment of silence at the beginning of each school day. I don't think your state constitution would offer more privacy rights than the federal Constitution for abortion, which is the big privacy dilemma coming up this month. But your point is interesting. The federal Constitution is, in effect, a baseline, or minimum.
States are free to go beyond it in protecting individual rights.
In fact, in recent disputes over whether certain homosexual acts between consenting adults are shielded from state criminal laws, state constitutions have been more protective of private sexual behavior than the U.S. Constitution. San Angelo, Tex. No, I don't think the government's defense of the Violence Against Women Act, at issue in the Morrison case, had anything to do with the solicitor general being denied time in the abortion case. My sense is that the majority of the justices didn't think the federal interest in the state abortion dispute was great enough to let the SG in. I also think the justices might have wanted to avoid further politicizing the dispute this election year.