This sounds like a scam a friend of a friend was caught in. He got the same sort of call and became worried about his computer. He gave them his credit card account numbers, bank acct details including passwords. He's a smart man but was worried about his computer and lost his head at the time. An hour or so later the penny dropped and he rang the police and the banks. Nothing was taken in the short amount of time but everyone concluded he'd been scammed. I think you did the right thing agony.
A friend of mine got the same sort of call as you did but her computer wasn't on at the time. She played along with them and asked if they could see what it was downloading at that precise moment. They told her they could see it was full of viruses and trojan horses. She told them it was interesting but nigh on impossible as her computer wasn't even turned on at the time.
As soon as an unsolicited call asks for any sort of personal details it's time to hang up. Or if you do think it might be legit and you might be interested, ask them for a number you can call them on. No phone number? Has to be a con. They give you a phone number? Check it in online. Any decent sort of company or organisation such as a charity will have a website now. You can ring them to ask if they have been soliciting information, donations etc.