Telephone calls from "Microsoft" and "Dell" technical assistance

I was recently hospitalized and many people were telling me what to do.  After I got home I received 2 calls informing me that my computer was sending serious distress messages to Microsoft about malware inside my hard drive that could do damage.  I allowed the "tech" to access my desktop, but got a visitor who interrupted the session.

I told my visitor how upset I felt at allowing someone I didn't know access my computer.  He told me to pull the modem plug.  That ended the session.

Today a "Dell" tech said my computer was sending Dell messages of distress and offered a session to access my computer to repair it.  When I ask, How do I know you are from Dell?  He replied that he had my service code and my email address.  I told him I'd decided to take my computer to have it serviced.  My visitor from yesterday suggested a business.

Does anyone know?  Does Microsoft or Dell legitimately call customers after messages from the Event Viewer? 

If I had not been ill, I don't think I would have believed such a call could be legitimate.