Thanks to @Schwern for the basic code. This version avoids copying in the 2 GB string by using substr directly. It supports the additional $position parameter like index. I also did the rindex version.

Call like index/rindex except call with a reference to to source string:

index64(\$string, $match);
index64(\$string, $match, $position);
rindex64(\$string, $match);
rindex64(\$string, $match, $position);


sub index64 {
    # Do index on strings > 2GB.
    # index in Perl < v5.22 does not work for > 2GB
    # Input:
    #   as index
    # Output:
    #   as index
    my $ref = shift;
    my $match = shift;
    my $pos = shift || 0;
    my $block_size = 2**31-1;
    my $strlen   = length($$ref);
    # No point in doing extra work if we don't need to.
    if($strlen < $block_size) {
        return index($$ref, $match, $pos);
    }

    my $matchlen = length($match);
    my $ret;
    my $offset = $pos;
    while($offset < $strlen) {
        $ret = index(
            substr($$ref, $offset, $block_size),
            $match, $pos-$offset);
        if($ret != -1) {
            return $ret + $offset;
        }
        $offset += ($block_size - $matchlen - 1);
    }
    return -1;
}

sub rindex64 {
    # Do rindex on strings > 2GB.
    # rindex in Perl < v5.22 does not work for > 2GB
    # Input:
    #   as rindex
    # Output:
    #   as rindex
    my $ref = shift;
    my $match = shift;
    my $pos = shift;
    my $block_size = 2**31-1;
    my $strlen = length($$ref);
    # Default: search from end
    $pos = defined $pos ? $pos : $strlen;
    # No point in doing extra work if we don't need to.
    if($strlen < $block_size) {
        return rindex($$ref, $match, $pos);
    }

    my $matchlen = length($match);
    my $ret;
    my $offset = $pos - $block_size + $matchlen;
    if($offset < 0) {
        # The offset is less than a $block_size
        # Set the $offset to 0 and
        # Adjust block_size accordingly
        $block_size = $block_size + $offset;
        $offset = 0;
    }
    while($offset >= 0) {
        $ret = rindex(
            substr($$ref, $offset, $block_size),
            $match);
        if($ret != -1) {
            return $ret + $offset;
        }
        $offset -= ($block_size - $matchlen - 1);
    }
    return -1;
}