I think it's normal to be anxious about loved-ones welfare, but this seems to be a problem or you wouldn't have put it on here. I agree with the others - this isn't right and you said it's not normal for you. I know you said you have no reason to be depressed, but depression isn't like that and, in my experience, it often brings anxiety with it. Maybe a trip to the doctor or even a chat with a friendly practice nurse might help?
The other thing is that trying not to think can be self-defeating. Maybe allow yourself a 'fretting' time, then when that time is up, do something else, but accepting that the worry is still there - just that you'll deal with it later.
Yes, I am speaking from experience and, yes, I know it's not easy. Hope it helps a bit though.