Originally, "Christmas Eve" meant the night before Christmas day.
Today we count days from midnight to midnight. That is, we begin each new day at midnight. On the ancient Jewish calendar, the day went from sunset to sunset. That is, sunset marked the beginning of a new day. When Christians borrowed holidays from the Jews, they borrowed this idea of the holiday starting at sunset. (See, e.g., http://www.catholicdoors.com/misc/christmaseve.htm)
Somewhere along the line we switched to the Roman practice of starting the day at midnight, but retained the "eve" of some holidays as the night before. So the night before Christmas day, that is, the night of December 24, is Christmas Eve. The night before New Years, that is, December 31, is New Years Eve. (That's the only holidays I can think of where we do this. Nobody talks about "Fourth of July Eve" or "Veterans Day Eve". Maybe there are other examples.)
Many people now use "Christmas Eve" to mean the entire day before Christmas and not just the night. From there it's a short step, I guess, to saying "Christmas Eve Eve" to mean December 23. But this is not accepted practice; it's more of a joke.
Such utterances are known as phatic. In the OED's definition, they 'serve to establish or maintain social relationships rather than to impart information, communicate ideas.' Exchanges about the weather, such as you describe, can made without a greeting such as 'Hello' or 'Good morning' and often occur between strangers.
Best Answer
"I hope you had a nice Christmas" or "I trust you had a nice Christmas" would both suffice. The latter might sound a bit overly-formal to some ears, but if it's a business contact you don't know outside of business that's not necessarily a bad thing.
Much is made in some quarters about whether it is better to refer to Christmas or the holidays generally, with differing opinions on both sides.
Personally as a non-Christian who does celebrate a different holiday near the same time of the year, I find nothing objectionable about people wishing me a happy Christmas - being happy at Christmas is definitely preferable to the alternative, after all.
Still, if you know that they celebrate Christmas or if they have mentioned Christmas previously as the reason they will be un-contactable for a while, then go with Christmas. If you know they celebrate another holiday (solstice and Chanukkah were both celebrated recently) then do mention the holiday they celebrated. If you're unsure then something like "I trust you enjoyed the break and are keen to start on the new year's projects" can avoid mentioning a particular holiday without sounding like you're avoiding it (there is a minority who consider anybody enjoying a holiday near Christmas that isn't Christmas to be a "war on Christmas" and get upset about avoiding being overly specific).
All that said though, "I hope you had a nice Christmas" can't go far wrong.
Also, if English is your second language and they know where you are from, a native greeting in your own language can be a nice touch, especially if context or similarity in etymology makes it easy to guess the meaning of.