There were several errors, but the main one was declaring a second @ten_array inside the loop. Others were:

1. Initialising @ten_array with 0 which doesn't clear the array as I guess you were thinking, but assigns 0 as the first element
2. Declaring square nested. It will not work as you expect
3. Assigning $number to @ten_array. Even if you hand't redeclared @ten_array that wouldn't have worked because it simply replaces any previous content of the array with a single element containing $number.
4. You use $number as a global inside square, but you call square passing a value into it. Usage and implementation are not consistent.

A cleaned up version of your code with a few other changes is shown below. Note in particular the use of strict (always use strictures) and a Perl for loop to count the loop iterations instead of hand roling a while loop to do that.

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

my $wanted = 3;
my @numbers;

print "Enter $wanted numbers.\n";

for (1 .. $wanted) {
    my $number = <>;

    chomp($number);
    push @numbers, $number;
    square($number);

}

print "The ten numbers were @numbers\n";


sub square {
    my ($number) = @_;
    my $square_num = $number * $number;

    print "$number squared is: $square_num\n";
}
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