String eval is considered a bad practice. Block eval is safe, and if you don't like it, use any of the try/catch modules available for a more mainstream look.

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

use Time::Piece;

sub validate_date {
    my ($date) = @_;
    return eval { $date eq Time::Piece->strptime($date, '%m.%d.%y')->s
+trftime('%m.%d.%y') }
}

use Test::More tests => 5;

ok   validate_date('02.28.19');
ok ! validate_date('02.32.19');
ok ! validate_date('02.29.19');
ok   validate_date('02.29.16');
ok ! validate_date('02.30.16');
[download]

The eq is needed because 02.29.19 is parsed without an error by Time::Piece, returning 03.01.19 when formatted back to a string.

Update: It passes Corion's test cases correctly.