Did I quit?

What is the name of your state? FL

I was asked to answer questions during an HR investigation and in one of my answers I mentioned wanting to put in my two weeks' notice. HR has decided to accept my resignation and said that I am not eligible for rehire because according to them I quit. I never formally put in my resignation can they say they've accepted it and that I quit?

Yes, they can.

It is generally wise not to mention to anyone at your workplace that you are thinking of giving your notice unless you are prepared to be immediately taken at your word.

Depending on the exact circumstances, you might be eligible for unemployment. However, no law requires that you formally resign or that they wait for a written resignation before they accept it.