Microsoft is a huge contributor to Linux and uses it heavily.

Linux also isn’t even close to a threat to windows for desktops.

They have no reason to at all and it would actually hurt them.

Well, to answer your question, if EEE was their plan, they are on step one. Step 2 would be to add a “killer” feature to linux, and make it proprietary. Given linux as a whole is pretty well fleshed out, that would be pretty difficult. So probably not gonna happen.