One way is to create two computed fields in your query, one of them using
the Day function on your date field, and the other using the Month function.
You'd then supply the appropriate values as criteria for those computed
fields. Note that you'd need to supply 1 for January or 6 for June using
this approach.
Another approach would be to use Format([MyDate], "mmmm d") as a single
computed field. Now you can supply the string "January 1" or "June 16".
However, you then need to be sure that they don't misspell the month name,
and that they don't accidentally put 2 spaces (or, in this case, January 01)