If your spouse still has their home country nationality, they will likely have to enter their home country using a passport issued by their home country. Many countries (including the U.S.) require their nationals who hold multiple citizenships to enter using the passport issued by that country, rather than a foreign passport.
I would plan for your spouse to travel with both their U.S. and home country passport. Exit and re-enter the U.S. with the U.S. passport, exit and re-enter spouse's home country with the home country passport.
Also keep in mind, as previous posters mentioned, some countries consider their nationals to have abandoned their citizenship of that country by acquiring the citizenship of a foreign country. For example, here in Uzbekistan, nationality law says that citizens of Uzbekistan are only recognized as citizens of Uzbekistan, even if they acquire another citizenship, and that acquiring foreign citizenship may be grounds for losing Uzbekistani citizenship. If you're willing to share your spouse's original nationality, someone may be able to advise you more accurately. It's certainly worth investigating and making sure you understand what ramifications, if any, acquiring U.S. citizenship will have on your spouse's home country citizenship.