The rainbow aura is limited to the Expedition Mode, and it requires reaching a certain spot on the map (Internally called EggSpot) as each of the slugcats. EggSpot is unmarked and is different for each of the slugcats, and it only spawns in Expedition Mode.

Once the EggSpot is reached, it is required to wait close to it for 200 ticks, or 5 seconds. The radius of the activation circle is 20 units (a single screen is approximately 1000 units in length). Once a sound effect plays (normally used for getting the Mark of Communication) and a circle appears, expanding from the slugcat, the EggSpot has been activated. It is required to shelter to preserve the EggSpot being reached.

To keep track of the completion, there is a visual indicator in the very bottom of the screen of the Challenges menu when starting an expedition, updating for each slugcat that you've reached the EggSpot with with their respective colour. For example, it appears like this once you've reached an EggSpot as every slugcat aside from Artificer:

The spots can be seen inside of the manual.

Each of the screenshots has been taken in the middle of the EggSpot being activated, and are ordered using the internal indexing. Note: Steam somehow broke the images in a previous version of the guide, I substituted them with the ones used in the Wiki. I also made the Wiki images, but they differ from the previous version by being 1366x768 instead of 1920x1020 (as the Wiki requires native resolution) and no longer showing the EggSpot activation.
Some of the locations may require expedition perks to reach, such as Rivulet's agility or Artificer's explosion jump. I'll try to explain the spot itself in case the images fail to load.

Survivor has EggSpot in room SU_B07 on coordinates (90, 629). This is located in the very right part of the Outskirts, in a horizontal room on the lower path to the batfly swarm room similar to an unlockable arena. To activate the EggSpot, climb up from the left entrance and stand on top of the small pole.

Monk has EggSpot in room SI_A06 on coordinates (250, 497). This is in the Communications Array, the room right below the very top. To activate the EggSpot, get on the pole hanging on the left and hang down from it.

Hunter has EggSpot in room LF_F02 on coordinates (1840, 962). This is in the Farm Arrays, in the giant open room right below the Sky Islands gate. To activate the EggSpot, get on a small floating platform in the right-middle screen.

Artificer has EggSpot in room LC_elevatorlower on coordinates (896, 385). This is in the Metropolis, the lower and darker room in bottom-right which is very vertical with 2 pair of poles on both sides. To activate the EggSpot, get to the very lower-right part of the room and stand on the small ledge there.

Gourmand has EggSpot in room VS_C04 on coordinates (2714, 579). This is in the Pipeyard, very close to the Industrial Complex entrance, in the open horizontal room nearby. To activate the EggSpot, get on the ledge in the top-right of the room and stand close to the wall, though not hugging it.

Spearmaster has EggSpot in room DM_ROOF01 on coordinates (5791, 599). This is in Looks to the Moon, the horizontal room on the top, close to Big Sis Moon herself, with multiple popcorn plants. To activate the EggSpot, get on the very top right corner.

Rivulet has EggSpot in room MS_bitteraerie6 on coordinates (2785, 183). This is in the Submerged Superstructure's Bitter Aerie (covered in snow), the horizontal room in top-right on the path to an antenna orb. For getting to the room, it is impossible to spawn in Submerged Superstructure and the gate to Bitter Aerie is still closed, so the only way (glitchless) is the same as in Story: by going through the Heart of the superstructure. You do not need any jetfish or bubbleweed to survive, as currents generated by the Heart (rarefaction cell is already inserted) will push you towards it.

To activate the EggSpot, get on the ledge in bottom-left and stand somewhat close by the pit, by the second vertical white line in the background. This may be the hardest spot, and keep in mind that you have to hibernate to save its activation. To exit the pit, a simple way is to pounce left and explosion jump directly up to get on the protruding ledge, then repeat the same to get onto the horizontal pole.

Saint has EggSpot in room UG_GUTTER03 on coordinates (1076, 1825). This is in Undergrowth, in the open room connected to the gate to Chimney Canopy. To activate the EggSpot, stand on a pipe visible in the right when near the pipe to the CC gate. The EggSpot is close to the wall.

As soon as you activate the 8th EggSpot, the Slugcat will activate the rainbow aura, rapidly changing between high-contrast colours. If you die before hibernating, the rainbow aura will be lost as the EggSpot deactivates, but re-activating it won't grant the aura back. Theoretically, it should persist after hibernating, but I happened to die that time, so I don't know whether it actually does, oops.

After activating all 8 of the EggSpots, the tracker in the bottom of the Challenges menu when starting an expedition will get replaced with a button, which allows to active the aura on any subsequent expeditions.
You cannot activate the aura for expeditions that have already started. The aura will also apply to slugpups and cyan lizards.

An example of a Slugcat with the aura:

When making this guide, I fully expected that Eggspedition was somewhat known, even if niche. According to Lee Moriya, the developer of the expedition mode, this is the first known discovery of the easter egg, and this is where I first documented it. It's amazing that secrets are still found almost a year after the DLC's release, and I wonder what else might be found in the future. Unfortunately, it wasn't the way it was intended to be discovered.

The day after I made this guide, I also created a section on the Expedition wiki page.[rainworld.miraheze.org] It's far more concise than the guide, though you are left on your own to figure out the room locations and coordinates with Dev Tools.

If you prefer a video guide, there are a few of them that I know of.

Lolight2 made I think the first one on YouTube: https://youtu.be/-ota800Smj0 The one by FuryForged is significantly shorter and relays most of the same information: https://youtu.be/MYNnu1iJsQw