Years ago (pre-BK) when I had an Amex Green card I once had problems paying as well; I personally found them very easy to work with - they really do want to help you find a solution.
Presumably you have a plan for paying them back, simply share it with them. In my case once I was caught up I was even able to have my account reactivated, which was a very pleasant surprise (it took me several months to get it cleared up). There is a deadline, though... I'm not sure what it is, but once you're past that the account will be closed and you'll 'officially' be in collections. If you think you'll be able to pay them back you might as well do it in such a way to keep the account if possible. 
I don't know your circumstances, but if you don't have a feasible plan for paying them back the phone calls aren't going to stop, and you'll need to explore your other legal options (again, depending on what you have going on).
I've been in the same situation while I was struggling to avoid filing BK. Most CC's will work with you as much as they reasonably can. At one point, I had two different cards accepting about 1/3 of the minimum payment as part of a payment plan to avoid going to collections.
Bottom line is they keep calling because they don't know what your intentions are. If you can't get help from family or a friend to pay the minimum due + fees, there are few options for you.
One thing you MIGHT be consider... If you have reason to expect you will be able to resume payments in the reasonably near future (30-60 days) you might consider doing a balance transfer to another card, even if you don't have any special offers. This is usually considered a payment by the source card, and would buy you a billing cycle or two, depending on timing. Don't transrfer a large amount... Just enough to cover maybe twice the minimum payment + late fees. This will keep you from getting a 60 day late, and them closing your card, possibly causing you MORE financial distress. Then tell them that you will be making a payment of $X in two weeks. This should reduce or eliminate the calls until the transfer posts.
Disclaimer: If you DON'T have a reasonable expectation of being able to resume payments normally any time soon, DON'T do this, as it will only increase your debt load. I offer the suggestion only as a temporary fix... I wouldn't do it more than once each way, which is why I say 30-60 days breathing room. Frequent balance transfers back and forth will undoubtedly raise red flags to both creditors.