Not unless you're looking to give yourself a reason to buy the next newest and most wonderful brake. Rubber seals in the two systems are designed to work with the specified fluid and will get destroyed in the presence of the incorrect fluid. Kiss any chance of a warranty buh-bye if you do try it, so DON'T DO IT.
If the syringes are clean and devoid of the other type of fluid... I see no issues.
But again... you don't even need a bleed kit for the Shimanos.
And most bleed kits are just a syringe and some clear rubber tubing. Go to a pet store/hardware store for the tubing and a pharmacy for the syringe. The Hayes uses a squeeze bottle... but I replace it with a syringe. Some bleed kits are specific to the brakes... for instance, fittings on the Hayes Nine are different than the fittings on a Juicy.