It can be harder sometimes to run in the cold and you could risk exhausting yourself out much sooner than you would when running in warmer weather.
But running in the cold weather is pretty healthy and will not make you sick unless you stay out in too cold of weather for too long then you risk frostbite or hypothermia if you don't wear gloves or protect your face.
You cannot catch the cold or flu from being out in the cold but it could increase your chances of catching Pneumonia so only run outside for a little while 30 to 40 minutes or less but no longer than an hour and always get back inside where it's warm and dress warmly for the cold weather and you should be fine.
I run a lot even in the cold weather but I dress warmly and don't stay out longer than 30 minutes at a time which is enough to give me some daily exercise so I stay fit and healthy but not long enough to cause me frostbite, hypothermia or any other harm.