Here's a way without external modules that I think does what you want. It does so by setting up a month to month num hash, and then after collecting all of the month names, get the unique ones, then sort them based on their month number in the $month_names hash:

use warnings;
use strict;

use Data::Dumper;

my %month_names = (
    January => 1,
    February => 2,
    March => 3,
    April => 4,
    May => 5,
    June => 6,
    July => 7,
    August => 8,
    September => 9,
    October => 10,
    November => 11,
    December => 12,
);

my @months;

while (<DATA>) {
    my ($client, $rest) = split /\|/;
    $rest =~ s/\s+//g;
    push @months, split /\d+/, $rest;
}

@months = uniq(@months);
@months = sort { $month_names{$a} <=> $month_names{$b} } @months;

print Dumper \@months;

sub uniq {
    my %seen;
    grep ! $seen{$_}++, @_;
}


__DATA__
IBM | February 1 March 5 July 4
Oracle| January 3 March 4 April 6 May 5
RedHat | March 2 June 3 August 1
[download]

Output:

$VAR1 = [
          'January',
          'February',
          'March',
          'April',
          'May',
          'June',
          'July',
          'August'
        ];
[download]