You can worry about ammonia and Phosphate, but I wouldn't worry about BBS carrying Ick.
You can keep them alive in old tank water with a gentle bubble flow. The diffuser is good for hatching them but to keep them a naked airline works best. The just need a gently flow, one bubble per few seconds is fine.
They will need food!
Algae paste, powdered eggs, yeast, anything in a powder and of nutritional value. Live Phyto is best as it actually cleans the water instead of degrade the water.
If you're not going to feed them then the refrigerator will slow down the metabolism and keep them alive but they will have absolutely no nutritional value for your Seahorse after 24 hours from hatching.