Use a state variable.

use warnings;
use strict;

my $input_name = 'DENIRO ROBERT';
my $print_flag = 0;
while (<DATA>) {
    chomp;
    if (/^[a-z]/i) {
        $print_flag = ($_ eq $input_name)
    }
    print "$_\n" if $print_flag;
}

__DATA__
DOE JOHN
123
DENIRO ROBERT
123
123
123
CUBE ICE
123
123
[download]

prints:

DENIRO ROBERT
123
123
123
[download]