There are many, many things logically, etymologically, and historically wrong with this argument, but I'll leave aside the issue of where the date of Dec 25 came from, when Christ was actually born, etc, and stick to etymology, philology, and phonology for this post. I have a PhD in this history-of-English and history-of-Christianity stuff, y'all - I am not making this up because I'm part of some pagan conspiracy to take over Christmas or anything. (I am not in fact a pagan, though people keep on assuming I am for some reason. I am a baptized and confirmed Catholic, and I went to Catholic school. As the Pope would not agree with a lot of my politics and practices and I don't agree with some of his, I am probably safest categorized as a "folk Catholic" for the sake of convenience; devotionally and philosophically I see the saints and spirits as being much closer and more involved in daily human activities than Bon Dieu and Jesus are - a sort of deism with animist/vitalist undercurrents as seen through a Franco-Haitian lens -- and I do not consider myself a pagan, and I do not divide the world into "Christian vs. pagan.")
Let's tackle "holiday." It shouldn't take much imagination to see that this word contains an early version of the word "holy." And in fact, it's original meaning did in fact refer to a specifically religious "holy day." The word comes from the Old English haligdæg "holy day; Sabbath," from halig "holy" + dæg "day." For instance: "giwarð haligdæg ongan in somnunga in sprece læra & monige giherdun giwundrade werun in larum his cweðende hwona ðas ðissum alle ðas is snytru hwelc gisald wæs him & mæhte ða ilcu ðaðe ðerh honda his gidoen bioðon" which translates "et facto sabbato coepit in synagoga docere et multi audientes admirabantur in doctrina eius dicentes unde haec omnia quae est sapientia quae data est ei et uirtutes tales quae per manus eius efficiuntur" in the Rushworth Gospels,  which is Mark 6:2: "When the Sabbath came, he began to teach in the synagogue, and many who heard him were amazed. "Where did this man get these things?" they asked. "What's this wisdom that has been given him, that he even does miracles!" (NIV translation). Here, the word translates "Sabbath." You could protest that it refers to Jewish rather than Christian holy days, but you'd be splitting inconsequential hairs and making a fool out of yourself, unless you reject Christian use of the word "Sabbath" in which case I'd like to know what version of the Bible you read.
Perhaps one more factor should be considered: "happy holidays" is a phrase that is naturally poetic to the English ear, and it would have been from Anglo-Saxon (Old English) days as well. The alliteration (or consonance, depending on whether you're a medieval or modern prosodist) of "h" sounds nice, and was the basis of Old English prosody (Old English poems did not rhyme; poetry was poetry in large part because of alliteration and medial breaks or caesura). Today, we still tend to like phrases like that; in fact, when we invent nonsense words or baby-talk in English, we tend to employ this same principle of repetitive primary consonant sounds. Phrases that stick with us and are pleasant to pronounce tend to last, and there are lots of them in English and their longevity is attested by the fact that many of them contain archaic words that don't occur outside these phrases much anymore (long live the Queen, kith and kin, practice makes perfect, tried and true, vim and vigor, best friends forever, insult to injury). So it's pretty silly to attribute the popularity of "happy holidays" as a phrase to some conspiracy to "take Christ out of Christmas." And it's also worth noting that not everybody at your workplace, in your contacts list, or in your neighborhood is Christian, so for some situations there is a respectful lack of specificity in using the term "holiday." And that should be noted too - a quick glance at the history of Christmas (and many other supposedly Christian holidays) will reveal that Christianity does not have a corner on and did not invent many of these holidays and traditions.
Now, as to xmas, if you've been reading this blog for a while, you may have noted that medieval scribes often abbreviated the hell out of things when writing in old manuscripts. If you could put yourself in the shoes of a 12-th century monk for a bit, and imagine being cramped over a wooden support holding up some vellum while you copied another bit of vellum in painstaking penmanship using the agonisingly slow pen-and-ink method in a cold, drafty scriptorium without much in the way of overhead lighting or ergonomic chairs, you might forgive a few abbreviations. If you can recall your own use of "etc," "i.e.," "PS.," "RSVP," "ATM," "Mrs.," "USA," "scuba," "Mr.," and "TV," you might be even more understanding. These things were simply common scribal practices, widely accepted and widely understood as conventional. Who would write stuff that nobody else could understand?
The bottom line is that "xmas" is an abbreviation of "christmas." The Greek letter Chi, written as X, is the first part of an ancient abbreviation used to signify Christ, an abbreviation that turned into a symbol or sigil. This is even in the dictionary, though the kind of rabid, slavering (or, equally numerous, sappy and sentimental though woefully uninformed) true-meaning-of-Christmas-defenders who complain about "xmas") probablay don't consult that book too much. But it's more fun to show you than to tell you, anyway. This can be seen in the Emperor Constantine's battle standard:
The thing at the top is "Chi Ro," a combined Chi (X) and Ro (P), thus the first two letters of the word "Christ." I haven't had time to personally chase down the earliest manuscript occurrence of it, but here's a post on ancient (as in Greek New Testament) abbreviations of holy names, aka nomina sacra, written by a Professor of New Testament Studies at Asbury Theological Seminary, which mentions the XP and many other common abbreviations in the earliest manuscripts of Christian and Jewish holy books. Many of these examples were not written in drafty winter conditions in Europe and England but rather in the much-warmer Mediterranean, so these abbreviations clearly have a long tradition and nothing to do with being sloppy or trying to "X Christ out of Christmas," for the love of all that's holy. Remember, there was no Kinko's, there was no printing press, and there was no office supply store where you just pick up another ream of paper for a few dollars. Vellum was not cheap, and preparing it was not quick. And these conditions continued to apply in the earliest printed works, such as the Gutenberg Bible, which is, I'd think, unarguably Christian; in it, abbreviations and contractions adapted from Latin manuscript usage continued to be used, because ligatures, abbreviations, and contractions helped save space, produce uniform and evenly spaced text and pages, and conserve paper. The Catholic Encyclopedia, which is published with a "nihil obstat" imprimatur (meaning it's as close to Church-approved as most anything gets), has a helpful, though long, page on accepted ecclesiastical and liturgical abbreviations. X for Christ is one of these.
My ultimate point here? It's silly, and not a little ignorant, to claim that the word "holiday" or the abbreviation "xmas" are pagan and/or cheapening Christmas and/or unChristian.
Now, many are complaining not about this etymology stuff but about commercialization and the mass hysteria that accompanies consumerism in some circles around the holiday season. That is also another post, and I am not trying to defend the pepper-spraying of rival shoppers or the trampling of one's neighbor in the pursuit of a Nintendo Wii; I consider that reprehensible (though I would like to point out that it's not just an "ugly American" tradition - mass frenzies and herd-mentalities are a pretty thoroughly human phenomenon, regardless of government, fiscal ideology, or religion; furthermore, some analyses of these events suggest that there is often more to it than a few people losing their damned minds over an iPod). I think people who would line up for this stuff at 3 am have skewed mentalities and need to get fulfilling hobbies and take a season off from the malls to give handmade gifts. However, I do support gift-giving, and would point out that gifts, in some form or another, have been important ways to establish and maintain bonds between human beings since before human beings had writing systems with which to record their traditions. I support sane, thoughtful gift giving, preferably done with an eye toward supporting independent artisans, small businesses, and local business. But if your child is too young to save for that insanely expensive electronic thingamajig herself, then she is too young to really appreciate what you're doing if you're standing in line at 3 am for it anyway. Get a hobby, folks. Learn a foreign language. Take up rock-climbing. Stay out of Wal-Mart.
BUT all this stuff has nothing to do with paganism or blasphemy or whatever, and that is my main point here. So when you use the words "holiday" and "xmas" to refer to the Christian holiday of Christmas, you are joining monks, priests, religious leaders, faithful, and holy people who have done so for over a thousand years.
Here's one Jewish take on the history of the Christmas holiday.
Here is a Christian take on how the ancient symbols of the Christmas holiday might be redefined/claimed for modern Christians.
Happy holidays! Buy local, stay out of Wal-Mart, support small sellers, local business, and cottage industries, emphasize quality over quantity, and teach your kids not to be greedy little shits by making them donate something to kids who don't even have a house to live in on Christmas, never mind a Wii and an iPad and the latest thingamajig. But don't get bent out of shape over the history of words or abbreviations if you're not going to take the time to actually get educated on the subject and on the history of your own religion and its holy texts.
 Skeat, W.W.The Four Gospels in Anglo-Saxon, Northumbrian, and Old Mercian Versions (Cambridge) [repr. Darmstadt 1970] .