777 means that any user on the system (with execute access for all the parent directories, anyway) can add anything to that directory. Web users are not system users, though, and most web servers (Apache included) won't let random clients write files there right out of the box. You'd have to specifically tell the server to allow that, and i'm fairly certain that's not what happened.
If you're allowing any file uploads, though, the upload folder needs to at least be writable by the web server's user (or the site's, if you're using something like suPHP). And if the web server can write to that directory, then any PHP code can write to that directory. You can't set permissions high enough to allow uploads and low enough to keep PHP code from running, short of making the directory write-only (which makes it pretty useless for fckeditor and such).
The compromise almost certainly happened because of a vulnerability in the site itself. Chances are, either there's a file upload script that's not properly checking where it's writing to, or a script that blindly accepts a name of something to include. Since the PHP code typically runs as the web server's user, it has write access to everything the web server has write access to. (It's also possible that someone got in via FTP, in which case you'd better change your passwords. But the chances of the web server being at fault are slim at best.)
As for what to do at this point, the best option is to wipe the site and restore from backup -- as has been mentioned a couple of times, once an attacker has gotten arbitrary code to run on your server, there's not a whole lot you can trust anymore. If you can't do that, at least find any files with recent modification times and delete them. (Exploits hardly ever go through that much trouble to cover their tracks.)
Either way, then set the permissions on any non-upload, non-temp, non-session directories -- and all the existing scripts -- to disallow writes, period...particularly by the web server. If the site's code runs as the same user that owns the files, you'll want to use 555 for directories and 444 for files; otherwise, you can probably get by with 755/644. (A web server would only be able to write those if it's horribly misconfigured, and a hosting company that incompetent would be out of business very quickly.)
Frankly, though, the "support guy" has the right idea -- i certainly wouldn't let a site go live on my servers knowing that it's going to be executing arbitrary code from strangers. (Even if it can't write anything to the local filesystem, it can still be used to launch an attack on other servers.) The best option for now is to remove all ability to upload files for now. It's obvious that someone has no idea how to handle file uploads securely, and now that someone out there knows you're vulnerable, chances are you'd keep getting hacked anyway til you find the hole and plug it.
As for what to look for...unfortunately, it's semi vague, as we're talking about concepts above the single-statement level. Look for any PHP scripts that either include
, require
, or write to file names derived in any way from $_GET, $_POST, or $_COOKIE.