Seeing as New Years at least appears to be a non-religious occasion and is a national, secular holiday, can I wish a coworker a Happy New Years?
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See this excellent write-up by Rabbi Michael Broyde, permitting it.
Rabbi Moshe Feinstein briefly remarks (when discussing Thanksgiving) "that it's more like New Year's Day" (and thus permissible). Though perhaps the origins may have been somewhat pagan, at this point those origins have long, long faded away in virtually all of society (how many churches have special New Year's Day services today?); New Year's is observed as a secular holiday today, and with observances that make practical sense. (People hope the new year is a good one.)
As Rabbi Broyde notes, this means you can wish people Happy New Year, and if you wish to do some contemplation on the passage of time, by all means. Drunken revelry and inappropriate behavior is inadvisable "be it on New Year's, Chanukah, or an ordinary Thursday night!"
The Wikipedia entry for New Year's Day indicates that it seems to have pagan origins. The article says, "This day is traditionally a religious feast, but since the 1900s has also become an occasion to celebrate the night of December 31, called New Year's Eve." Someone born on the 25th December would have his bris on 1 January (Wikipedia and here).
It would seem therefore to depend whether you are wishing a happy new year (=12 months, secular) or a happy New Year's Day (= one day, religious).