I see nothing wrong with saying "Happy Holidays". As others have said, it is a more inclusive greeting for the season, as it includes not only Christmas, but New Years, Hanukkah, Kwanzaa, Festivus, the Solstice, etc.,. It is not a cop-out to say "Happy Holidays". It is a recognition of reality that a lot is going on in December and various people are celebrating various holidays.
I wonder how certain Christians would react if I wished them a "Happy Hanukkah". I can think of a few places where I've lived where (very sadly) it would NOT get a kind reaction: "What makes you think I'm a Jew?" Some "less than Christian" Christians I have known in my lifetime would definitely be offended, I am sure. Wishing them a Happy Kwanzaa would REALLY not go over well....
There are jerks on either side making this a mess. While in college, each year we organized a group to go caroling. One grad student sniffed indignantly when I invited her to join us and then went on to lecture me on how Christmas was all a silly myth perpetuated by weak-minded people. *Great* Well, the rest of us (several of various Christian denominations, a few atheists, a few agnostics, and a couple of Jews) went out and had fun singing and wishing people well in song. We also got invited in at several houses for snacks & warm drinks (OK- and for some "Christmas Spirits", too).
Just as bad (if not downright evil and manipulating) are the folks who claim that saying "Happy Holidays" is sign of a war on Christmas. Please give me and the rest of the country a break. Many of us who say "Happy Holidays" do not hate Christmas or the Baby Jesus or Christianity or religion in general. We are just being more inclusive in our well-wishing. Also, if I know someone Christian, I will say "Merry Christmas", like when I was a kid in the little town where there were only Christians (well, and probably a few closeted atheists).
I just wish people would forget all the hastily-filled-out cards, and stressful joyless frantic shopping excursions and the gaudy lights and going in to debt for gifts that no one actually needs. Celebrate the true meaning of Christmas by showing a little compassion and understanding and reflect on your life and be thankful for what you have.
The Grinch learned his lesson. Why can't we? The Whos would have been just fine if their presents had been dumped off Mount Krumpet. Would we?
.... Sadly, I doubt it!