Direct indexing of function return value in Fortran

No, that is not possible in Fortran. You could, however, alter your function to take an additional index array that determines which elements are returned. This example illustrates this possibility using an interface to allow for an optional specification of the indices (simplified greatly thanks to the comment by IanH):

module test_mod
  implicit none

  contains

  function squareOpt( arr, idx ) result(res)
    real, intent(in)              :: arr(:)
    integer, intent(in), optional :: idx(:)
    real,allocatable              :: res( : )
    real                          :: res_( size(arr) )
    integer                       :: stat

    ! Calculate as before
    res_ = arr*arr

    if ( present(idx) ) then
      ! Take the sub-set    
      allocate( res(size(idx)), stat=stat )
      if ( stat /= 0 ) stop 'Cannot allocate memory!'

      res = res_(idx)
    else
      ! Take the the whole array    
      allocate( res(size(arr)), stat=stat )
      if ( stat /= 0 ) stop 'Cannot allocate memory!'

      res = res_
    endif

  end function
end module

program test
  use test_mod
  implicit none

  real    :: arr(4)
  integer :: idx(2)

  arr = [ 1., 2., 3., 4. ]
  idx = [ 2, 3]

  print *, 'w/o indices',squareOpt(arr)
  print *, 'w/  indices',squareOpt(arr, idx)
end program

No.

But if it bothers you, you can write your own user defined functions and operators to achieve a similar outcome without having to store the result of the function reference in a separate variable.