Código PHP:
$cadena='[{"id":"1","title":"Josue Ismael","allDay":"true","start":"2010-07-02 00:00:00","end":"2010-07-02 00:30:00"},{"id":"2","title":"Josue Ismael","allDay":"false","start":"2010-07-02 16:00:00","end":"2010-07-02 18:30:00"},{"id":"3","title":"prueba ","allDay":"false","start":"2010-07-16 08:00:00","end":"2010-07-16 14:30:00"},{"id":"4","title":"prueba ","allDay":"true","start":"2010-07-09 19:00:00","end":"2010-07-09 19:30:00"},{"id":"5","title":"prueba ","allDay":"true","start":"2010-07-09 19:00:00","end":"2010-07-09 19:30:00"},{"id":"6","title":"prueba ","allDay":"true","start":"2010-07-16 00:00:00","end":"2010-07-16 00:30:00"},{"id":"7","title":"prueba ","allDay":"false","start":"2010-07-16 08:00:00","end":"2010-07-16 14:30:00"},{"id":"8","title":"prueba ","allDay":"true","start":"2010-07-09 19:00:00","end":"2010-07-09 19:30:00"},{"id":"9","title":"Josue Ismael","allDay":"false","start":"2010-07-01 19:00:00","end":"2010-07-01 19:30:00"}]';
$cadena=str_replace('"false"', 'false', $cadena);
si además querés quitarle comillas a los "true" también:
Código PHP:
$cadena=str_replace(array('"false"', '"true"'), array('false', 'true'), $cadena);