The single biggest issue for Christian Nationalists may be the use of the generic greeting "Happy Holidays" over the Christmas-specific greeting "Merry Christmas." I don't think that a couple of years ago anyone was saying that it was designed to undermine Christianity or even that it excluded Christmas and Christians. Today, though, people like Lou Dobbs claim that saying Happy Holidays is a deliberate attempt to exclude Christians.
Why has the phrase "Happy Holidays" become popular? Over a period of many years Christmas has become less of a Christian-centric holiday and more of a generic cultural holiday. It is celebrated by many non-Christians in various ways and the religious connotations are lost even on some Christians themselves. Christmas isn't very "Christian" anymore.
More important, however, is the fact that America has become more religiously diverse and there are more religions celebrating holidays around the end of the year. Adherents of these religions expect the same respect and consideration that Christians want for themselves; saying "Happy Holidays" as a generic greeting means treating everyone equally rather than showing favoritism towards just the Christians who celebrate Christmas. This is why even the White House under George W. Bush tends to wish the best during the "Holiday Season" rather than more narrowly wish people a Merry Christmas.
"Happy Holidays" is thus the most appropriate greeting during the holiday season when you don't know for sure what a person is celebrating. Even if you do know that they celebrate Christmas or Hanukkah, though, "Happy Holidays" may be as good as if not better than "Merry Christmas" or "Happy Hanukkah" because they may be celebrating more than one holiday -- at the very least there is New Year's and many families are multi-religious. By choosing a narrower greeting, you're only wishing them well for one of their holidays. Why not wish them well for all the holidays they celebrate?
The above image is based upon a World War II poster designed to frighten people about what might happen in America if the Nazis weren't defeated. This seems an appropriate theme for this topic because the rhetoric of Christian Nationalists indicates that they consider attempts to deny them control over the public square to be much the same as Nazi repression. This image reflects Christian Nationalists own fears back on them: the fear that they will be persecuted merely for expressing their Christian beliefs.