What I've settle on is shown below. It won't remove a single blank line at the beginning of the file, but I don't need that. It may not be the most efficient way to do this, but it make sense to me. I learned a bunch. Thanks!
#!/usr/bin/perl
while (<>) {
$last = $this;
$this = $next;
$next = $_;
$lastblank = ($last =~ (/^\s*$/));
$thisblank = ($this =~ (/^\s*$/));
$nextblank = ($next =~ (/^\s*$/));
print $this unless (! $lastblank) && ($thisblank) && (! $nextblank);
}
print $next unless ($nextblank) && (! $thisblank);