If someone chooses to say "happy holidays" rather than "merry Christmas" out of sensitivity or political correctness, that does not constitute cancelling. The term cancelling is usually used for deliberate and sustained social pressure and boycotting campaigns. As far as I am aware this has not occurred on any significant scale to try to stop people from saying "merry Christmas".
In the other direction, however, this has very much happened. Conservative Christian organizations have called for boycotts of companies who don't mention Christmas in their Christmas marketing (or, in some cases, even just for mentioning non-Christian holidays) repeatedly over the last two decades. [0]
To avoid weak-manning here: I'm sure this is a pretty fringe group and not at all representative of Christians in general. But it is clear that the campaign to force people to use traditional Christmas greetings and imagery has always been more prominent and aggressive than the campaign to abolish traditional Christmas greetings.
[0] https://en.wikipedia.org/wiki/Christmas_controversies#Retail...