You might want to check out URI::Find. That way, you can use it to search your text for links, then grep the resulting list of valid URLs for the eid= part, printing out any that match.

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

our $textfile = "rfl.txt";
our ($text, @urls, @eidurls, $finder);

use URI::Find;

# Create the finder object with a callback to add
# URLs to our list.  Note that URI::Find expects
# the callback to return the (possibly modified)
# URL to be reinserted into the text (hence the
# trailing shift).
$finder = URI::Find->new( sub { push @urls, $_[0]; shift} );

# Read the file into $text, using "slurp mode"
# (ie, the whole file is read in at
#  once instead of a line at a time)
{
  open my $textfh, " $textfile" or die "Can't open file: $!\n";
  local $/; # slurp mode
  $text = <$textfh>;
  close $textfh;
}

$finder->find(\$text);          # Find 'em.  They'll be in @urls now.
@eidurls = grep /eid=/, @urls;  # Find the ones we're interested in.

print "The URLs are:\n";
print " $_\n" for @eidurls[1..$#eidurls]; # skip the first one
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bbfu
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Nor do the wind, the sun, and the rain.
We can be like they are.