The concept of EVERYONE group was introduced in Maximo v7. This brought a whole host of functionality changes which are described in detail here. Luckily though, it only brought with it one new configuration option—the ability to change what group is treated as the EVERYONE group. The reasons you would want to do this are fairly limited. They are almost entirely limited to the desire to use the group name EVERYONE for another purpose in Maximo.
To make this change, head over to the Security Groups application. In the Selection Action menu, click Security Controls.

This brings up the Security Controls dialogue. From here, you can enter in or select the name of the group you wish to act as the EVERYONE group. Once the OK button is clicked, the change is saved.

Beware, though, the security group is not automatically applied to existing users. One must manually add the group to all existing users. Also, the previous EVERYONE group will be left as part of all existing user security profiles. This means that if your new EVERYONE group has different privileges granted, and you do not remove your old EVERYONE group, your users could potentially have access elevated beyond what they should. Therefore, if you are simply replacing the EVERYONE group for all users, you have to remove the old EVERYONE group and add the new EVERYONE group for each individual user.