Not a good idea. Storing vital data like session expiry and user name entirely on client side is too dangerous IMO, encrypted or not. Even if the concept is technically safe in itself (I can't answer that in depth, I'm no encryption expert), a break-in could be facilitated without compromising your server, just by acquiring your encryption key.
Somebody who gets hold of the key could generate session cookies at will, impersonating any user for any length of time, something the classical session concept is designed to prevent.
There are better and scalable solutions for this problem. Why not, for instance, set up a central session verification instance that all associated servers and services can poll? Look around on the web, I am 100% sure there are ready-made solutions addressing your needs.
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