What. Why do you think that even a possibility? Goji is a King. Eggman isn't even really a king; he won't be one until we conquer the Green Hills. This will suck for Goji and weaken him, but he will recover.
And no, most citizens won't love us. Most of them are apathetic about Goji, believing that he is better than everyone else. Living under the rule of the Eggman Empire means living as one of Eggman's subjects. It's not a great life! Sure, we'll abolish the fun police, but they're still going to be expected to provide blood and toil for the glory of Eggman. Hell, the easiest thing for us to conquer Tokyo-To will probably involve just kicking out Goji and putting a puppet of our own in command.
They aren't going to rejoice about joining the Eggman Empire.