In the recursive example, if a malicious user provides sufficient amounts of bad input, Perl may kill your program with a "deep recursion" message. (The number 4000 comes to mind as the recursion limit, but I'd have to check.)

In the goto example, you've basically written a while loop, which is how most people would do it.

I'd remove the duplicate code from the end of each if/else statement, and write it as such:

#!/usr/bin/perl -w
use strict;

# &subname can have unpleasant side effects
a();

sub a {
    my $a;
    while (lc($a) ne 'q') {
        chomp($a = <STDIN>);
        last unless $a;

        if ($a =~ /\D/) {
            print "Not an integer!\n";
        } else {
            print "You typed: ->$a<-\n";
        }
    }
}
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There's more than one way to do it.