I too had this problem. Read the entire thread with a lot of useless suggestions (mostly from QB).

Just got this to work -- and was very glad I didn't have to manually enter 303 transactions.

I went to 

Backup a Local Copy of my QB

Help | Update Quickbooks Desktop

Went to the "Update Now" tab.

Clicked "Get Updates"

Did this a couple of times until things were all updated with the current date / time 

Ran the update.

Restarted QB - got logged in.

Tried again -- Same 304 error.

One of the suggestions in this thread said to disconnect the feed and reconnect.

So I unlinked the account

Then set up the bank feed again 

And it WORKED!

Not sure if all the updates helped with this endeavor or just the reconnection to the American Express bank feed did the trick.

Phew.