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Kelvin is completely correct.
Here are the two relevant clips from the NFHS Rules Book. You will see that while there are some restrictions on re-entry, the only time that an injured player is barred from returning is when that player was determined to be unconscious, then a doctor's permission is needed.
Rule 8 - SECTION 2 ATTEMPTING PERSONAL-FOUL FREE THROWS
The free throw(s) awarded because of a personal foul shall be attempted by the offended player. If such player must withdraw because of an injury or disqualification, his/her substitute shall attempt the throw(s) unless no substitute is available, in which case any teammate may attempt the throw(s) as selected by the team captain or head coach.
Rule 3 - SECTION 3 SUBSTITUTION
ART. 1 . . . A substitute who desires to enter shall report to the scorer, giving his/her number.
a. Between quarters, at halftime and during a time-out, the substitute must report or be in position to report to the scorer, prior to the warning signal which is sounded 15 seconds before the end of the intermission for a 60-second time-out and prior to the 20-second warning during a 30-second time-out.
b. Substitutions between halves may be made by the substitute or a team representative.
c. During multiple free-throw personal fouls, substitutions may be made only before the final attempt in the sequence and after the final attempt has been converted.
EXCEPTION: When a player is required by rule to be replaced prior to administering the free throw(s), then all other substitutes who have legally reported may also enter the game.
d. If entry is at any time other than between quarters, and a substitute who is entitled and ready to enter reports to the scorer, the scorer shall use a sounding device or game horn, if, or as soon as, the ball is dead and the clock is stopped.
e. A captain may request a defensive match-up if three or more substitutes from the same team enter during an opportunity to substitute.
ART. 2 . . . The substitute shall remain outside the boundary until an official beckons, whereupon he/she shall enter immediately. If the ball is about to become live, the beckoning signal should be withheld. The entering substitute shall not replace a designated jumper or a free thrower except as in 8-2 and 3. If the substitute enters to replace a player who must jump or attempt a free throw, he/she shall withdraw until the next opportunity to substitute.
ART. 3 . . . A substitute becomes a player when he/she legally enters the court. If entry is not legal, the substitute becomes a player when the ball becomes live. A player becomes bench personnel after his/her substitute becomes a player or after notification of the coach following his/her disqualification.
ART. 4 . . . A player who has been replaced, or directed to leave the game shall not re-enter before the next opportunity to substitute after the clock has been started properly following his/her replacement.
ART. 5 . . . A player who has been injured to the extent that the coach or any other bench personnel is beckoned and/or comes onto the court shall be directed to leave the game, unless a time-out is requested by, and granted to, his/her team and the situation can be corrected by the end of the time-out.
ART. 6 . . . A player who is bleeding, has an open wound, has an excessive amount of blood on his/her uniform, or has blood on his/her person, shall be directed to leave the game, unless a time-out is requested by, and granted to, his/her team and the situation can be corrected by the end of the time-out.
NOTE: (Arts. 5, 6) If players from both teams are directed to leave the game because of injury/blood, both teams must request and be granted a time-out in order to keep each player in the game.
ART. 7 . . . A player who has been determined apparently unconscious shall not return to play in the game without written authorization from a physician (MD/DO).
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