Mongoose: Find, modify, save

The user parameter of your callback is an array with find. Use findOne instead of find when querying for a single instance.

User.findOne({username: oldUsername}, function (err, user) {
    user.username = newUser.username;
    user.password = newUser.password;
    user.rights = newUser.rights;

    user.save(function (err) {
        if(err) {
            console.error('ERROR!');
        }
    });
});

Why not use Model.update? After all you're not using the found user for anything else than to update it's properties:

User.update({username: oldUsername}, {
    username: newUser.username, 
    password: newUser.password, 
    rights: newUser.rights
}, function(err, numberAffected, rawResponse) {
   //handle it
})

findOne, modify fields & save

User.findOne({username: oldUsername})
  .then(user => {
    user.username = newUser.username;
    user.password = newUser.password;
    user.rights = newUser.rights;

    user.markModified('username');
    user.markModified('password');
    user.markModified('rights');

    user.save(err => console.log(err));
});

OR findOneAndUpdate

User.findOneAndUpdate({username: oldUsername}, {$set: { username: newUser.username, user: newUser.password, user:newUser.rights;}}, {new: true}, (err, doc) => {
    if (err) {
        console.log("Something wrong when updating data!");
    }
    console.log(doc);
});

Also see updateOne