You really do need to fix that tank. That is one of the worst cases of algae taking over a tank I've seen.
As for the egg thing, if they lay eggs behind or in contact with the anemone they will not survive. For them to survive you would have to remove them from the tank and you won't be able to remove them if they are attached to the tanks glass. I've also never seen clown fish lay eggs that close to a anemone, I don't believe the eggs have the protective coating that makes them immune to stings.
Also that anemone is on its way out of this world. It's very bleached, and I wouldn't expect it to last more than a few weeks longer.
Arrrrrgh, clean the sand!!!!!