There's really no way you can catch diseases from the cold weather itself.
You can however catch diseases from others being cooped up indoors during the cold weather.
Contrary to popular belief it's not the cold weather that actually makes you sick. You get sick when you catch a virus and catching a flu or cold virus just happens to happen more often in the winter months.
You can get hypothermia though if you're out in too cold of weather for too long and you might catch pneumonia from being too cold but you have to catch a virus to be sick.