I had the same phone call.  It cited Microsoft windows errors.  I let them into my computer with me watching.  I am wondering how to look for an evil program that they may have attached.

Make sure they didn't turn off your antivirus and firewall. Run the antivirus program. Also go to filehippo and download maleware bytes and spybot search and destroy (you want to make sure it is updated first thing). Run them.  If you don't have a antivirus program, Avast is top notch.  All are free.

Back it up by running one of the free virus scans online like Norton.  Then change all your passwords for everything.  Maybe they might not have downloaded a virus, but they had access to any and all info on your computer.

I have talked to this person before.  Always uses a different name.  Says he is from Microsoft and my computer is in a very serious condition of viruses, etc.  Called from 1-000-000-0000 Out of Area.  Said he is Antony from New York.  I asked for his number to have my husband call him back (yeah right).  He gave me this number 347-305-0007.  
Talked to what sounded like him a few months ago and he eventually got very nasty and angry when he realized I was not going to pay for anything!  Dirty mouth and yelling in phone. We are on DO NOT CALL LIST but we still get these unwanted calls..

NIce guyssssssss.

It's nice compony........