Keith and Roger would make a mess of it.
This generally means Keith and Roger are working together, and they would make a mess of it.
Either Keith or Roger would make a mess of it.
Either-or would generally mean one of them would make a mess of it and the other wouldn't (and this doesn't specify which). But that is a strange thing to say, so without further context, most people would probably interpret this as: Keith would make a mess of it and Roger would make a mess of it.
Keith or Roger would make a mess of it.
This generally means either (a) one of them would make a mess of it and the other wouldn't or (b) Keith would make a mess of it and Roger would make a mess of it.
Both Keith and Roger would make a mess of it.
This generally means Keith would make a mess of it and Roger would make a mess of it. So this is probably the best one to go with for what you want to say.
Either of Keith or Roger would make a mess of it.
This is a slightly unusual phrasing, but it would also mean Keith would make a mess of it and Roger would make a mess of it.
"Either" roughly means "one of them", whereas "either of" means "it doesn't matter which one".
The more natural version of the above would use "them" to refer to Keith and Roger, e.g.:
Keith or Roger? Either of them would make a mess of it.
"or" translates to "and" in some examples above, because it's basically saying pick one of two (which is "or"), and the one you picked would make a mess of it (which means they would both make a mess of it, which is "and").