There are unfortunately several issues with your code:

• You're using the second capture group as the filename, which according to the regex is actually the third word in the line. You probably meant something like /^zone\s+(\w+)\W+\w+\s*$/, and then you can use the capture variable $1.
• As jcb mentioned, you've got a stray > in the regex.
• sprintf($outfh'.txt',... is not valid syntax. Please make sure to always provide a runnable SSCCE when asking questions.
• I don't understand why you have a $filecount variable, since according to your description, you want the filenames to be derived from just the section names.
• Using the variable $outfh to hold both the filename and the filehandle is not good practice, you should use a separate variable to hold the filename.
• You print to the output file unconditionally, even when it's not open yet. The same goes for the close.
• You say you want the output files for this example input to contain six lines each, but you don't say which of the seven input lines.

If I fix these issues (keeping $filecount), I get:

use warnings;
use strict;

my $infn = 'testdat.dat';
open(my $infh, '<', $infn) or die "$infn: $!";

my $outfh;
my $filecount = 0;
while ( my $line = <$infh> ) {
    if ( $line =~ /^zone\s+(\w+)\W+\w+\s*$/ ) {
        close $outfh if $outfh;
        my $outfn = sprintf '%s-%d.txt', $1, ++$filecount;
        open($outfh, '>', $outfn) or die "$outfn: $!";
    }
    if ($outfh) {
        print {$outfh} $line or die "print: $!";
    }
}

close($outfh) if $outfh;
close($infh);
[download]

However, this produces output files that are 7 lines or longer, because they include the endoffile marker and any lines following it. If you want the endoffile to be excluded from the output, the fix is fairly simple:

    if ( $line =~ /^zone\s+(\w+)\W+\w+\s*$/ ) {
        ...
    }
    elsif ( $line =~ /^endoffile$/ ) {
        close $outfh;
        $outfh = undef;
    }
[download]

If you want something more complex than this, then it'd be better to switch to a state machine type approach, which I showed some templates for at the top of this node.

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