You need Delayed - it represents an expression whose evaluation is delayed until its value is externally requested. I used seconds so you can see the numbers are different every time you access the page.
CloudDeploy[Delayed[Column[{
Style[ToString[AccountingForm[Round[DateDifference[Now,{2017,1,1},"Second"]]]]<>
" seconds to New Year",20,FontFamily->"American Typewriter"],
Style[ToString[AccountingForm[Round[DateDifference[Now,{2016,12,25},"Second"]]]]<>
" seconds to Christmas",20,FontFamily->"American Typewriter"]
}]]]