In a calendar I saw this in reference to the Children's Day:
Happy Children's Day!
I guess 六一 is a shorter form of 6月1日.
Is that form of writing dates only used for special dates? How widespread is that usage?
This form of abbreviation is only reserved for special dates. Some popular Chinese holidays take this abbreviated form:
Any date can potentially become "special" following a major event or incident that occurred on that date or for a short period of time surrounding the said date. The abbreviated form is, therefore, used to identify the event:
This is predicated on the fact that the listener is fully aware of both the date and the event in question. The abbreviated form would be meaningless otherwise.
A dot, "·", may be added between the month and the day in writing, especially when there is a possible ambiguity:
The reading stays the same, however.