Here's one way to solve it:

Store every day a user is scheduled to work. With a column to state if this is holiday or not, e.g.:

create table employee_work (
  employee_id integer
    not null,
  working_day date
    not null
    check ( working_day = trunc ( working_day ) ),
  is_holiday  integer
    default 0
    not null
    check ( is_holiday in ( 0, 1 ) ),
  primary key (
    employee_id, working_day
  )
);

Load it up with every employee's working days as needed.

With this in place, when they request holiday run:

update employees
set    is_holiday = 1
where  employee_id = :emp
and    working_day between to_date ( :start_dt, <fmt> ) 
                   and to_date ( :end_dt, <fmt> );

And if they want to cancel days within a holiday period, set those days back to "not holiday" in the range:

update employees
set    is_holiday = 0
where  employee_id = :emp
and    working_day between to_date ( :start_dt, <fmt> ) 
                   and to_date ( :end_dt, <fmt> );

This avoids the need to do any manipulation of the existing date ranges.

If you want to show someone which days they have booked holiday, run:

select * from employees
where  employee_id = :emp
and    is_holiday = 1