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Billy, I don't think that the "leaving the court for an unauthorized reason" rule was designed to prevent players from leaving the court UNintentionally. Rather, it was designed to prevent players from leaving the court intentionally to gain an illegal advantage, to show disgust and/or resentment to the officials, or for other reasons not specified (perhaps to prevent a Malice in the Palace situation from occurring in a HS game) . Only intentional leaving of the court is disallowed in NCAA (the violation is for leaving the court of one's own volition), so it would not be a stretch to assume that the NFHS rule is to be enforced in a similar manner, barring a contrary case play or official interpretation from the NFHS.
In your situation (a player getting sick on the court), the covering official would stop play immediately, unless the opponents of the injured/sick player have an advantage and immediately use it), leaving the court for an unauthorized reason would not apply, because there are other procedures covering that situation. In my previous answer, I emphasized that a player might not have many valid and legal reasons for leaving the court, because a team member may go to change his jersey outside the visual confines of the court, an injured team member can leave the court for treatment, etc.
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