Your code has a number of errors in it. Consider the following:

#!/usr/bin/perl
use strict;

my $i;
my $data;
open(DATA," basicdez.dat") || die "Cannot open datfile: $!\n";

# this entire loop is pointless.  its almost never a good 
# idea to read an entire file into a single string, unless 
# you are going to be spitting it right back out.

while(<DATA>){
   chomp;
   unless ($_=~ /"EOS"/){
      $data .= "$_";
   } elsif ($i ne 1) {
      $data .="\n";
   }
   if ($_=~ /"EOS"/){
      $i=1;
   }
}
close(DATA);

my @data=split(/\n/, $data);

# i find it odd that although you name every element of your
# list, you don't name the loop variable.  Either do one, or
# both.  Be consistant!
for(@data){
   my($first_name,$last_name,$address,$city,$state,$phone)=split;

   print "$first_name,$last_name\n";
   print "$address\n";
   print "$city, $state\n";
   print "$phone \n";
}
# not needed
exit;
[download]

Heres how I would have done what you wanted (with explanations)

#!/usr/bin/perl -w # use warnings!
use strict;

# suck the entire file into an array, rather than just 
# putting it into a file.
open(DATA," basicdez.dat") || die "Cannot open datfile: $!\n";
my @data = <DATA>;
close(DATA);

# remove your eof element.  If you can help it, take it out of the 
# dat file completely, so you can remove this line of code.
@data = @data[0..@data-1];

# name and scope your loop variable
for my $data (@data)
{
   # remove those pesky quotes.  If you can help it, take those out
   # of the dat file too!  If i remember correctly, they are
   # needed in vb; however, Perl isn't a weak language like
   # vb is.  It can handle raw text ;)  You should delimit 
   # your text with a single obscure character (the standard
   # obscure character (the standard for flat dbs is the
   # pipe | symbol.)
   # However, given your data as it is now, this will work:
   $data =~ s/" "/""/g;
   $data =~ s/""/|/g;
   $data =~ tr/"//d;

   # split on the newly created pipe delimited entry.
   my($first_name,$last_name,$address,$city,$state,$phone)
   = split /\|/, $data;
   print "$first_name,$last_name\n";
   print "$address\n";
   print "$city, $state\n";
   print "$phone \n";
}
[download]

There you have it. I hope I was clear enough; if I wasn't, please reply. Good luck!

Update: fixed tr and split, thanks to projekt21