Helen's blue eggs

Just because eggs are blue doesn't mean they were laid by a bluebird. Bluebirds are cavity nesters, and VERY rarely lay their eggs outside of a cavity. (When desperate, they may use a ledge.)
Some people who report finding bluebird eggs in odd locations (in a potted plant for example) are really seeing eggs of another species, like the American Robin nest, eggs, and nestlings in these photos.
House Finches lay eggs that are bluish-green, and sometimes use a nestbox. Starlings also lay blue eggs, but they are bigger than bluebird eggs. About 4-5% of bluebirds actually lay white eggs. Also thrashers lay blue eggs.

Arizonia birds...

The most common bird to lay blue eggs is the American Robin. However, there are other birds native to Arizona that lay blue/bluish eggs. A few of these are the Crissal Thrasher (blue-green egg), Western Bluebird (Pale Blue egg), Mountain Bluebird (pale blue), Townsend's Solitaire (Pale blue to whitish), Hermit Thrush (pale blue to bluish green), Red-winged Blackbird (Pale blue-green), Scott's Oriole (pale bluish white), Western Tanager (pale blue or bluish green). there are others.