Can I say "I am twenty years young."
At least in America, saying "I am X years young" is a very informal, and normally somewhat humorous, way of saying "I am X years old." Its usage nowadays is typically restricted to address the age of someone, usually in their 40s or older, while attempting to "dance around" (in a joking manner) the fact that they are as old as they are. I think it is a good example of the prevalence of "youth culture" in America.
Strictly speaking, it is incorrect to say ". . .years young." Always use ". . .years old," especially when the situation is even the slightest bit formal.
As a real-world example, when I was a kid in elementary school, it was common for the administrators to announce a teacher's birthday over intercom. They would always say "Mr/s. [teacher] turned X years young today!" The reason it is supposed to be slightly jocular is because people often don't want to get any older than they already are.
How old — I mean young — are those who are said to be "X years young"?
I searched The Corpus of Contemporary American English for * years young, then went through the entire list to convert words to numbers, fix errors, and remove false positives. I also removed two outliers, 200 and 14 billion. (Note: not all of these hits refer to people!) Then, I graphed it as a histogram:
The mean is 66.6 years: Most often this expression is said humorously of older people, conveying that despite their age they shouldn't be considered "old". It is chill, informal, casual.
However, there are quite a few examples where it's someone really young being talked about. For example, the Washington Post:
"And now — at 14 years young — Hunter Lussi!" brayed race director Bob Vigorito, who greeted each finisher over a loudspeaker.
Raw numbers (unordered):
80, 80, 80, 80, 53, 53, 53, 53, 40, 40, 40, 40, 86, 86, 86, 86, 92, 92, 92, 92, 100, 100, 74, 74, 74, 37, 37, 37, 28, 28, 50, 50, 38, 38, 91, 91, 90, 90, 88, 88, 87, 87, 83, 83, 82, 82, 81, 85, 85, 80, 80, 94, 94, 129, 125, 68, 72, 71, 70, 17, 40, 51, 99, 97, 95, 37, 64, 47, 125, 52, 83, 84, 78, 77, 76, 75, 52, 51, 72, 67, 64, 63, 62, 59, 58, 57, 68, 55, 54, 75, 49, 43, 35, 32, 31, 26, 25, 22, 21, 20, 17, 14, 128, 106, 100, 10, 53