#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

my $Map_Request_Date  = 'Jan 1 2017 12:00';
my $Map_Due_Date      = 'Jan 31 2017 12:00';
my $Map_Cutover_Date  = 'Feb 28 2017 23:59';
my $Map_Complete_Date = 'Mar 1 2017 12:01';
my $Map_Approved_Date = 'Dec 31 1999 0:01';

my @Variables = ($Map_Request_Date, $Map_Due_Date, $Map_Cutover_Date,
                 $Map_Complete_Date, $Map_Approved_Date);

my $X = 0;
for my $Date_Ref (@Variables) {
    say $Date_Ref;
    $Date_Ref =~ s/ +/ /; #When day is a single digit it creates two w
+hite spaces
    my ($Month, $Day, $Year, $Time) = split / /, $Date_Ref, 4;
    my %Months = ( Jan => '01',  Feb => '02',  Mar => '03', Apr => '04
+',
                   May => '05',  Jun => '06',  Jul => '07', Aug => '08
+',
                   Sep => '09',  Oct => '10',  Nov => '11', Dec => '12
+'  );

    $Day = "0$Day" if 1 == length $Day;
    $Variables[$X++] = "$Year-$Months{$Month}-$Day";
}
say for @Variables;
[download]

for (same as foreach ) aliases the values it iterates over, so you don't need the array at all:

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

my $Map_Request_Date  = 'Jan 1 2017 12:00';
my $Map_Due_Date      = 'Jan 31 2017 12:00';
my $Map_Cutover_Date  = 'Feb 28 2017 23:59';
my $Map_Complete_Date = 'Mar 1 2017 12:01';
my $Map_Approved_Date = 'Dec 31 1999 0:01';

for my $Date_Ref ($Map_Request_Date, $Map_Due_Date, $Map_Cutover_Date,
                  $Map_Complete_Date, $Map_Approved_Date
) {
    say $Date_Ref;
    $Date_Ref =~ s/ +/ /;
    my ($Month, $Day, $Year, $Time) = split / /, $Date_Ref, 4;
    my %Months = ( Jan => '01',  Feb => '02',  Mar => '03', Apr => '04
+',
                   May => '05',  Jun => '06',  Jul => '07', Aug => '08
+',
                   Sep => '09',  Oct => '10',  Nov => '11', Dec => '12
+'  );

    $Day = sprintf '%02d', $Day;
    $Date_Ref = "$Year-$Months{$Month}-$Day";
}
say for $Map_Request_Date, $Map_Due_Date, $Map_Cutover_Date,
        $Map_Complete_Date, $Map_Approved_Date;
[download]

Note the usage of sprintf to format the day.

But it's even easier when you use Time::Piece :

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

use Time::Piece;

my $Map_Request_Date  = 'Jan  1 2017 12:00';
my $Map_Due_Date      = 'Jan 31 2017 12:00';
my $Map_Cutover_Date  = 'Feb 28 2017 23:59';
my $Map_Complete_Date = 'Mar  1 2017 12:01';
my $Map_Approved_Date = 'Dec 31 1999 0:01';

for my $Date_Ref ($Map_Request_Date, $Map_Due_Date, $Map_Cutover_Date,
                  $Map_Complete_Date, $Map_Approved_Date
) {
    say $Date_Ref;
    my $tp = 'Time::Piece'->strptime($Date_Ref, '%b %d %Y %H:%M');
    $Date_Ref = $tp->ymd;
}
say for $Map_Request_Date, $Map_Due_Date, $Map_Cutover_Date,
        $Map_Complete_Date, $Map_Approved_Date;
[download]

($q=q:Sq=~/;[c](.)(.)/;chr(-||-|5+lengthSq)`"S|oS2"`map{chr |+ord
}map{substrSq`S_+|`|}3E|-|`7**2-3:)=~y+S|`+$1,++print+eval$q,q,a,
[download]