If you use the DATE type to store your dates in your mysql database then they will be stored:
So to convert M-dd, yy to say, store something in the correct Mysql you'd use PHP, something basically along the lines of :
$name = "William Hunting" ;
$date = date('Y-m-d', strtotime('Sep 13, 10')) ;
// build an insert query
"insert into yourtable startDay, personName values '$date', '$name'" ;
Not testing for failure, or illegal dates, which youd need to do.
Note the quoting of the values in my simple example which assumes you are escaping the values yourself for security reasons and not using PDO or prepared statements.
Having said that, tinker with your JQuery date picker and see if you cannot just format the return value to be YYYY-MM-DD and save this extra step - or look for one which offers that option.