My preference would be to explicitly test for compilation failure with $@.

use warnings;
use strict;

$| = 1;
print "Input regexp: ";
my $user_input = <STDIN>;

my $regex = eval { qr/$user_input/x };
if ($@) {
    print "Your regexp would not compile. The error was:\n$@\n";
    exit 1;
}

my $string = 'abcdef';
print "$string - " . (($string =~ $regex) ? "matches\n" 
                                          : "no match found\n");
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