They're called sponge anemones, in the same family as light bulb anemones and aiptasia's, though the sponge and light bulbs aren't nearly as invasive.
They'll reproduce if well fed but won't spread out and take over like aiptasia's, I'm not sure if a berghia would eat them or not, I had peppermint shrimp that ate aiptasia's but never touched the lightbulb and sponge anemones that were also in the tank, berghia might eat them but I have no experience with berghias.
-Edit:
If you want to get rid of them, kalk mix may work, I've used a mix of 1pt kalk and 1pt anhydrous calcium chloride pellets (prestone driveway heat or kent's turbo pro calcium) to 1pt water, mixes hot since the calcium chloride reacts with the water, then target delivered with a syringe or feeder.
This has worked wonders against aiptasias in the past for me and was effective against sponge and lightbulb anemones that were encroaching on corals.
Edited by aquablue - September 30 2012 at 2:21pm