I'm a breeder of American budgies (hopefully, I'll move on to the elusive English budgies). If I were you, and I didn't want my hen to egg-bind, I sure as heck would leave the eggs in there. Taking away eggs just encourages her to produce more eggs, and it taxes on her calcium and really poops the hen out (no pun intended). Seeing that you don't want chicks, you can addle the eggs. It's rather controversial, however, because some bird fanciers think of addling as "birdie abortion". *shrugs* I don't see it that way, but you might think differently.

Anyhow, my breeder [person, not bird] has a few older pairs that have bonded and she cannot separate them. To discourage egg-laying, she covers the cage for 14 hours and allows minimal sunshine. Just enough so that the birds get their Vit D and all, but not enough to encourage breeding. She also houses them in fairly quiet areas during breeding season. Something to keep in mind...