You can't. You need to use the mod Dino storage v2 https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=1609138312&searchtext=

Cryopods will keep the level, stats, and color but nothing else. You put a Tardis one in a cryopod and it will become a normal but blue creature. The soul balls in this mod allows you to keep the special attributes of your dinos and keep them safe in a ball. If you rename it though it will not keep it's type in the name. Like you can name your skeletal raptor something but in the description of the animal it will just say raptor. I use dino storage because I just like it better. It's better cryopods. You get a paintball gum to shoot the balls from to get your critter from a distance, you make them from crystal, and best of all, no cryosickness and they don't expire if you don't charge them.

I have a cryopod mod. My question was, what kind of additional data do I need to save so that my cryopods work with shiny dinos.

All of Shiny's special abilities are buff based, so if your modded cryopods will retain persistent buffs, that should do the trick. Colours and extra levels are assigned when dinos are selected by Shiny, and are completely vanilla.

fantastic! thank you.

Im using the dino storage. Yet when tamed their special effects disappear. Also TamedShowSparkles=False is set to True. So any idea why their not retaining there sparkles/special effects? Also would colorful dinos plugin effect shinies at all?