Try a calculation field which converts FM Format into MySQL (apparent) format, such as:
Year (FMDate) & "-" & Month (FMDate) & "-" & Day (FMDate)
and search on that field instead?
Wonder if that MySQL field is a text field instead of a date field. If not, you might want to research the internal storage specifications for a MySQL date field. Most systems store dates as serial number that counts the number of days from an arbitrary, early date to the date displayed. In FileMaker, the minimum possible date is: 1/1/0001--which has a numeric value of just 1. (1 day after 12 / 31 / 0000).