I had my e gift card balance of 1 K totally wiped out. It had been used by some thief even while I was at another Hyatt. It was replaced but then when I went to use the replacement card it also was down to zero just as I was checking out of Hyatt Waikiki last month.
We had just started our 3 week trip but a quick call and a new e card was sent. I quickly used it in Kauai before it too might be compromised.
So paper is no safer than plastic, but Hyatt quickly replaces your balance. I guess it's still worth the 10% savings, just disconcerting.