So, to summarize...
You couldn't pay your credit card bill last Feb so you called Amex to let them know. Instead of living up to the monster-lender stereotype, they were pleasant and helped you out by setting you up on a "care program" which I assume is some sort of reduced minimum or deferred payment plan.
You then missed payments and as such, did not complete the alternative payment plan they offered. Then, you ignored billing statements that indicated your balance was overdue.
Now, though the balance was not paid within the agreed schedule and is still owed, you went out shopping for a mortgage to buy a second home and discovered that Amex is not helping you out by assisting you in maintaining a high credit score.
Basically, not only are you unable to imagine why Amex is negatively reporting you, but you actually feel that you're the one being ripped off in this situation?
I see...
Here's the key to keep this from happening in the future. Pay at least the minimum balance due each month to avoid late & missed payment flags from hitting your credit report. If the company you owe is nice enough to offer alternative payment arrangements during a period of financial difficulty and you accept said arrangements, then live up to those arrangements instead of shopping around for second home mortgages. Finally, if you fail to pay said debt to the agreed schedule, try to understand that a negative mark on your credit is the added price you pay along with interest and fees. It is to be expected.
There are many legitimate reports around here of people getting ripped off and IMO, this is not one of them.