The Eurostar trains may well be unable to operate outside the laws of Physics. However, very cold weather in northern France (and indeed the UK) is not unusual at this time of year. It has occurred on and off for as long as I can remember (which is a long time) and has nothing to do with climate change, whether natural or man-made. Similarly not unusual is the problem of large temperature changes occurring as trains enter (comparatively) warm tunnels.
The Italians and in particular the Swiss have been operating trans-Alpine train services for a few more years than Eurostar has been operating under the Channel. I think the temperatures outside the Alpine tunnels are probably lower and the snow levels higher than those this weekend around Folkestone and Calais. Despite this I do not recall any wholesale train failures occurring in Alpine areas because of these reasons.
One or two things spring to mind:
When designing their trains Eurostar may have adopted the approach that local authorities and railway companies in England do when planning for cold weather. (i.e. it does not happen often enough for us to spend any significant money on it, so we’ll just let the poor sods get on with it for a few days – as they did last year, and the year before that, and the year before that).
Or, they may have taken more seriously than I did the advice given to people in southern England a few years ago. This suggested they should redesign their gardens with arid conditions in mind by planting various breeds of cactus, and move towards plants such as grapes and bougainvillea which are more suited to the sub-tropical climate which we are, without doubt, about to enjoy.