Yes, for in-store purchases Costco will only accept American Express credit cards. The key word is credit card though because they do accept debit cards as well. If you purchase from the Costco website however they accept all major credit cards.
I highly recommend applying for the Costco TrueEarnings Card (link to review & official application) which acts as a Costco Membership and gives you 3% cash back on gas, 2% cash back on travel and dining, and 1% back on everything else including other purchases at Costco.
We shop at Costco a good amount and that has saved us some serious $$$.