I'm intrigued by the post title - there's not a regex in sight.

$hash->{key}=="string" is performing a numerical comparison between a hash value (presumably a string) and a string - did you mean $hash->{key} eq "string" thus performing a string comparison ??

Using strictures would have helped - consider the tidied up code (so that it compiles!):

use warnings;
use strict;

my ($str, $logoutLink);
my $Session = {
    usrSystem   => 'MFR',
    usrLevel    => 'some_val',
};

sub foo {
if ($Session->{'usrSystem'} eq "MFR" && $Session->{'usrLevel'}=="!M")
      {
      $str .= "<leftIndex>";
      $str .= xmlFileData("Content/SSResources", $Session->{'usrSystem
+'}.".xml", "mLeftIndex()");      
      $str .= xmlFileData("Content/SSResources", $Session->{'usrSystem
+'}.$Session->{'usrAccount'}.".xml", "mLeftIndex()");
      $str .= $logoutLink;
      $str .= "</leftIndex>";
      return $str;
      }
###Shows regular for everyone except this access level... ?


    if ($Session->{'usrSystem'} eq "MFR" && !$Session->{'usrLevel'}=="
+!A")
   {
      $str .= "<leftIndex>";
      $str .= xmlFileData("Content/SSResources", $Session->{'usrSystem
+'}.".xml", "mLeftIndex()");      
      $str .= xmlFileData("Content/SSResources", $Session->{'usrSystem
+'}.$Session->{'usrAccount'}.".xml", "mLeftIndex()");
      $str .= $logoutLink;
      $str .= "</leftIndex>";
    }
    return $str;
}

foo();
[download]

$ perl $_
Argument "!M" isn't numeric in numeric eq (==) at tst.pl line 11.
Argument "some_val" isn't numeric in numeric eq (==) at tst.pl line 11
+.
Undefined subroutine &main::xmlFileData called at tst.pl line 14.
[download]

Ignoring the undefined sub error, you can see that the compiler would have told (or at least given a clear indication to) you what was wrong.