What is the connect between Christian holy days and pagan holidays?

A few friends of mine are under the impression that many so-called Christian days of worship or celebration are actually pagan feasts poorly concealed behind the guise of genuine Godly worship, and that recognizing these "fake" occasions would actually be a betrayal of the true God to Whom they were "mockingly meant to give praise". The rest of my friends are under the impression that these days of worship were deliberately placed on days which previously marked pagan holidays to emphasize that conversion had been made from the ungodly to the Godly behaviours, kind of like a purposeful replacement (ref. spanish conquistadors a la Cortes). So my question is this - WHO'S RIGHT? Am I insulting God by celebrating things on days which possibly had marked pagan, or heavenforbid demonic, events? And there are more false gods than there are days in the year so isn't it kind of pointless to try and "choose" the days to celebrate most things? Like where would you put Christmas???