Answer
No, Al-Husayn ibn al-Qasim did not serve under the Rashidun Caliphate, but rather the Umayyad Caliphate. The Rashidun Caliphate, which lasted from 632 to 661, was succeeded by the Umayyad Caliphate, which lasted from 661 to 750. Al-Husayn ibn al-Qasim served as a general and governor in the Umayyad Caliphate, and is particularly known for his military campaigns in North Africa and Spain.