You do have to use the JetBlue card to get the free bag benefit,and, unfortunately, the benefit is only for the primary card holder.

"Free First Checked Bag
You and up to 3 travel companions on the same reservation who have purchased Blue fares each receive their first checked bag free on JetBlue-operated flights. All ticket purchases (original and exchanges) must be made with the JetBlue Business Card Account and your TrueBlue membership account number must be entered at the time of booking. "

(Elsewhere in the T&C, they define 'you' as primary card holder.)