Jim McKelvey and his former intern Jack Dorsey got the idea for financial-technology company Square Inc. the last time a global crisis sent the economy into a nosedive.

It was 2009, and Mr. McKelvey—a glassblower, computer scientist and serial entrepreneur—had lost a sale of one of his artworks because he couldn’t accept American Express cards. Though neither he nor Mr. Dorsey, now CEO of Square and Twitter Inc., knew much about the world of credit-card transactions, his frustration inspired the creation of Square’s signature…

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