Well, I don't think so:
1.) It's not so difficult to get the UniqueID out of the HTTP header (when it's set), therefor you can get the cookie value and try to detect the algorithm. That means there is not such a big difference concerning security compared to the get variable.
2.) Normally the UniqueID with its creation algorithm is not the key to your site, but it's a reference to your temporary login information in a db or something. That means that being capable of creating cookies does not mean you can access the site. Only if there's a db entry with your login details corresponding to that cookie, you get access.
3.) What might be a problem is fetching someone else's cookie, transmitting it to site b and get access with the other persons account. However you could save an IP address together with the cookie details, that would make it harder to hijack.
ciao,
Matthias