OK I experimented a little with eof, and I found eof (no handle, no parens) returns true on the last line. So If I extend the end side of the flipflop operator with an extra test on eof, and set a flag to true only if it succeeds because of this extra test, then I can safely check for the last match, too. I do need to modify the regex a little to allow for this, too.

#! perl -w
my $buffer;
LINE: while(<DATA>) {
    my $eof;  # flase by default
    if(my $counter = (/^name\s.*\{/ ... (/^name\s.*\{/ || ($eof = eof)
+))) {
        if($counter == 1) {    # begin
            $buffer = '';
        }
        $buffer .= $_;
        if($counter =~ /E/) {  # end
            if($buffer =~ /name(.*?)(^name|\z)/ms) {
                print "Found: $1\n";
            }
            redo LINE unless $eof;   # retry with same line, unless en
+d condition is BECAUSE OF eof
        }
    }
}
__DATA__
name value {
   test 1;
   test 2;
   test 5;
}
name three {
   eat 4;
   eat 5;
}
name four {
   eat 6;
   eat 7;
}
[download]

Result:

Found:  value {
   test 1;
   test 2;
   test 5;
}

Found:  three {
   eat 4;
   eat 5;
}

Found:  four {
   eat 6;
   eat 7;
}
[download]