I didn't have time to read the other replies you received but I was having the same issues several weeks ago. After WEEKS of not being able to connect AMEX, HOURS on the phone, and QB support unable to resolve the issue... I finally figured it out.  Hope this helps!  (BTW... I'm copy and pasting from my community post)

For anyone having the same issues as me.  After getting in touch with someone who finally knew what he was doing with QBO, he found and resolved the issue I've been dealing with for the past 3 weeks.

FOR ACCOUNTANTS

If you're having trouble logging into AMEX for a client, go to YOUR business QB home page (not theirs) and click on the APPS tab on the lefthand side of your screen.  Next, select the "Clients Apps" tab at the top of the screen.  Under the client you're having issues for, look for the American Express App.  Click the AMEX app.  See who the app was "Connected by".  In my case, it said it was connected by: Others which meant my client tried to connect it, before contacting me.  Once I found that out, I had to use her QB login and password to log into her company account (as a user and not as an accountant).  I then followed the steps listed below and BOOM!  We're back up and running again!

FOR BUSINESSES

If you're having trouble connecting your AMEX bank feed to QB, click on APPS in the toolbar to the left of your page.  Look for the American Express App.  On the AMEX app, select the drop down menu on the ACTIONS button and then select launch.  Follow the instructions and you're good to go!

For AMEX, the difference between the "Business Card-OPEN" option and the "Personal or Corporate Card" option when connecting the account is this:

• The Business Card-OPEN is actually an app (I didn't know this) so your transactions download automatically.  You must enroll to use this free option.  Clicking the "Update" tab in the banking screen will not download additional charges/payments in the AMEX bank feed.
• With the Personal or Corporate Card option, you must click the "Update" button to download new charges/payments.