For shutdown I have a few circuits monitoring the inputs/outputs to check if we're getting backed up or running dry on either of the fruits. If that's the case the input belts are stopped, egg->egg duplicator inserters are disabled and the chests that hold the one egg being recirculated go into trash mode.

There is a separate unit that constantly ensures there are at least a couple eggs avaliable and "requests" (buffer chest) any excess and is surrounded by turrets.

During startup (science below limit and ingedients on input belts) the spoilage->nutrient inserter is enabled which priority feeds just the first three chambers that make flux. Once any flux has been made (>0 flux on the belt || flux->nutrient machine working) the spoilage inserter stops. That's enough for the flux->nutrient machine to start and the whole system becomes self sustaining.
As part of startup a requester chest requests 1 egg, the rest of the egg duplicators are setup in such a way that they propagate down the line (a direct inserter between them is enabled if the recirculating chest for the next chamber is empty and the egg->sci inserter gets disabled). Once there are eggs in the recirculating chests the startup is considered complete and the requester chests turn off.

There's also a chest that pulls from the main nutrient line to make sure there's always loads of spoilage ready for a restart.