Ah the most wonderful time of the year when the true snowflakes, who are not only completely unaware of irony, but unaware of actual facts and history.
The word holiday is derived from the old English word meaning 'Holy Day'. So, they have nothing to worry about, the person wishing them a happy holiday, any time of the year, is being seriously religious.
Added to that, it is not hard to comprehend that they are just using a shortened version of Merry Christmas and a Happy New Year, encompassing the 2 in one very innocent phrase.
Do they get this riled up if somebody says to them 'Have a nice long weekend' instead of 'Happy Easter'?
My favourite is the video of Trump doing the annual President message and saying something like 'I am the first president to say Christmas in a long time'. Cue the montage of every single president for the last 20+years saying the word Christmas.