By the looks of it, you are almost there. Apart from the two identical lexical variables problem hardburn mentioned, all you need is getting the remote file:

my $remote_file="remote_file";
my $openstring="/usr/bin/rexec $host -l $user cat $remote_file |";
# note the cat command, and the pipe at the end
open FILE, $remote_file or die "Blerch: $!\n";
$md5=Digest::MD5->new();
while (<FILE>) {
  $md5->add($_);
}
close FILE;
print $md5->b64digest;
warn "Code is untested.";
[download]

Of course this presumes your .rhosts or hosts.equiv files are set up correctly.

Update: Fixed typo in code (replaced [ by { )

CU
Robartes-