In that case the syntax will be tricky. You will need to see which string functions apply to your database's SQL.
for featureclasses in oracle you can use INSTR to find the position of the dash, SUBSTR to get only the part before the dash, TO_DATE to convert it to date and finally you can do '<', '>' operations with the dates in the where clause.
it works out to something similar to this in Oracle
where = "TO_DATE( SUBSTR(DateUserField,1, INSTR(DateUserField, '-') -1), 'YYYYMMDD') < TO_DATE('23-06-2011', 'DD-MM-YYYY') AND TO_DATE( SUBSTR(DateUserField, INSTR(DateUserField, 1, '-') -1), 'YYYYMMDD') > TO_DATE('22-06-2011', 'DD-MM-YYYY')"
For file geodatabase or shapefile, I don't know if you can do the string functions.