I am running into basically the same issue. A student went in and was working on the assignment in the Online Text box. When he finished the assignment, he hit the submit button and it sent him back to the login page. He contacted me to find out if I had received the assignment, and I hadn't.
Then I went in to check my settings. Everything seems right in the backend of that assignment, but when I went to look at the actual assignment it had the same message about "no submissions required" and there was no option to answer online or submit a file--yet the settings were set to allow submissions. And, previously I had been able to do so in testing it.
I did the fix mentioned above in this thread of turning off the allow submissions and then turning it back on, and that seemed to work. He was able to submit his assignment and so was another student. BUT when I went back into the backend of that assignment again after they did, the settings have changed! ??? I am not sure what is happening or how to keep that from happening.
Thanks for the help so far! Any more help would be greatly appreciated!