Entrepreneur Christopher Burch has been a loyal user of American Express Co. cards since 1979. This year, he switched almost all his spending to J.P. Morgan Chase & Co.’s Sapphire Reserve card.

Regaining customers like Mr. Burch, who made the Forbes list of wealthiest Americans in 2014, is one of the big challenges for incoming AmEx Chairman and Chief Executive Stephen Squeri. Named successor to longtime chief Kenneth Chenault on Wednesday, Mr. Squeri takes over as the card giant stresses it is building momentum after...