for some reason, if you tear some grass (a big handful) and put it on the crown board, the ants vacate. They may move back in when it dries out, but just repeat.
I usually scrape all the eggs off - Cheap, easy and using things to hand.
The spaces in a hive that bees cannot access make favoured places for ants cos they are dry and warm. Keeping the ants out is a never ending battle if you want to fight it, me, I just give up and live and let live. That's unless they are argentine ants in that case I exterminate.
Here's a tip though, sometimes I'll get a hive just so bad with ants I need to do something. In that case I use the gas burner that I keep in the truck and burn them, I leave the burned ants on the hive mat, and other ants never come back. Or not for a very long time at least.