Do I have to pay back a sign on bonus after a firing if there was no contract stating I have to ?

I received a sign-on bonus (said it was a sign on bonus in an e-mail with no stipulations, pay stub refers to it as a bonus, and I never signed any contract and wouldn't have in order to receive the bonus) if I was fired with no notice that I was in danger of being fired, after 2 1/2 months am I obligated to pay it back? After I was notified I was being let go, they began to start referring to it as a "pay advance"

I was interviewed as a project manager, but my offer letter listed me as a "Project Director", and then they changed my title to "Director of Operations" after a week or two on the job. The company was as sketchy as they come, and after the rest of the employees returned from a European project, while I worked their other three U.S. projects when they were gone, they let me go stating that I did not measure up to the expected performance. I had never been a titled project manager, and they bumped me to Director. There was only one other full time employee, so I wasn't really directing anything. I did implement all of their projects to a new Project Management software while also performing direct physical labor as more of a lead technician, which was something we discussed would not happen. It was one of the strangest employment experiences I have ever had. I worked for my previous company for 8 years, and 6 years at the one before that, and know what being a good employee involves.