Really? I can reproduce the OP's results on WinXP and Linux.
WinXP, Perl 5.8.8, ActivePerl 820
>perl -e"print qq{print <<ENDHERE\nabc\nENDHERE}" > test.pl
>perl test.pl
Can't find string terminator "ENDHERE" anywhere before EOF at test.pl
+line 1.
>perl -e"print qq{print <<ENDHERE\nabc\nENDHERE\n}" > test.pl
>perl test.pl
abc
>
Linux, Perl 5.8.4
$ perl -e'print qq{print <<ENDHERE\nabc\nENDHERE}' > test.pl
$ perl test.pl
Can't find string terminator "ENDHERE" anywhere before EOF at test.pl
+line 1.
$ perl -e'print qq{print <<ENDHERE\nabc\nENDHERE\n}' > test.pl
$ perl test.pl
abc
$
If I were to guess, I'd say the parser looks for <"\nENDHERE\n"> instead of <"\nENDHERE" followed by either "\n" or end of file>, and it does so because it's easier and because every line is suppose to end with a line feed in unix.