Will my budgie lay eggs?

This depends on the female, but yes. Usually, the male checks out the nest, if he is pleased, he will show it to the female, and she will decide if it's up to her standards to lay eggs in. Many times they may be hesitant to lay eggs when only nest materials are provided without a nest, since the eggs will be exposed and it isn't their nature. However, a female comfortable with her atmosphere may have no issue laying eggs without a nest box, when proper materials are available. Building a nest is not part of their wild habitat since they usually nest in hollow trees ect, so nest building isn't seen as often in budgies, and often times will effect breeding behavior.

Immature females also lay eggs, often times infertile, on the bottom, and have lower breeding standards. They aren't concerned very much in all of the details needed to properly rear their young before mating. In more experienced females, nesting behavior is part of what triggers the release of breeding hormones, and may require it to breed. Adding a nest encourages mating, however not providing one may discourage it to an extent, but it doesn't mean they won't breed.

yes. but it will may a great harm for your bird. female can effected by egg binding. so be careful dear.