No it should be only be done by license Doctor.
The schools should not be getting into something that they can and will be liable for.
It will open the door for all kinds of law suits.
I can recognize that schools feel somewhat responsible for the sexual behavior of their charges. However, it seems to me that giving out pregnancy tests isthe wrong intervention done too late. Much better to resist the pregnancy early than to detect it late. It would be far more appropriate to give out contraceptives than pregnancy tests. Pregnancy will out in any case.
However, I think that the schools are not choosing based upon effectiveness but on what they think will produce the least community resistance.
This thread suggests the difficulty of their decision.