If an airline agent tells you that, they are incorrectly telling you their airline's policy. As far as I know, no US carrier requires a US passport from US citizens to board a flight to Puerto Rico or most if not all other US territories. (For some reason I vaguely recall hearing of an odd issue with Guam.)
From Delta's website:
Traveling to a U.S. Territory
If you are a U. S. citizen and want to travel to one of its territories, you will not need a U. S. Passport. Sufficient ID would be a U. S. driver's license or other valid photo identification, the same as if you were flying to New York or Seattle. However, U.S. residents with foreign passports would be subject to the same entry procedures in one of the territories that they would be on the mainland.
From United's website:
U.S. Territory Travel Document Requirements
Passengers 18 years of age or older are required to have a valid, current U.S. federal or state-issued photo ID that contains name, date of birth, gender, expiration date and a tamper-resistant feature for travel to or from a U.S. Territory (Guam, Puerto Rico and U.S. Virgin Islands).