I think recent cold/flu symptoms is all they want to see. I have been fighting a respiratory virus for several weeks. Some symptoms are similar to COVID - cough, labored breathing - but unlike typical COVID symptoms my fever has been low, and I haven't had body aches or chills. Nevertheless, they wanted to test me. I didn't even ask for it. They suggested that I do it. Also, whatever I have, I picked up from a family member who, in turn, picked it up traveling in Thailand and Japan in late January - early February. So that may have been part of why the provider wanted to test me.
Edit: the only caveat is that to be eligible for the antibody test, your symptoms must have started at least a week or two ago. I don't know what the cutoff is, but I think it is something like 10 days. So even if you have really severe symptoms, if you have only had them for a couple of days, you would not be eligible for the antibody test.