I think the biggest thing I would have approached differently (other than using a CPAN module to do date work) is to divide the problem into parts and have code to perform each part.

# input:
chomp(my $date = <>);
# parse, return numeric year,month,day or empty list on error
if (my ($y,$m,$d) = split_date($date)) {
  # print out in Month dd yy format
  print "Date is: ", format_date($y,$m,$d), "\n";
} else {
  print "invalid date: $date; please try again\n";
}
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It doesn't make as much sense to have part of your conversion done in the main code and part done in a subroutine. Ditto for printing an error message in the main code or the successful formatting in a sub.

Update: make comment agree with code re: order of returned values; said parsing but meant conversion