As many have said in the comments, it is a terrible idea for many reasons. The first and foremost among them is that the vast majority of content is dynamic, and user specific, and it wouldn't make sense to cache anyway.
Furthermore, your cache would inherently lack the ability to determine sensitive information from non-sensitive information [edit] if a website is misconfigured. [/edit]
Imagine that I'm User A and I visit my email inbox at a hypothetical address https://www.qwertyuiop-mail.com/inbox
Their server has identified me as having logged in, and assumes a secure connection, and shows all the messages to me.
User B decides to check his email also at https://www.qwertyuiop-mail.com/inbox and because he just went directly there, your cache server says, "Hey! I have this page cached! I'll just serve User B what I just served User A" and bam, User B now sees User A's page, and probably a sizable portion of what only User A should see.
As far as the server owner is concerned, his system should work just fine because he isn't expecting anything to be cached. You will have effectively created a passive man-in-the-middle attack that any user on your system could potentially abuse.
If you want to do this kind of thing, I would recommend only doing it for certain white-listed sites that you know contain no login anything, or for which you prohibit logging in in some way. (Wikipedia may be a good candidate for this as long as you block login to the site)