Is a “blue bird” the same as a “bluebird”？

Is “blue bird” in the following quotation from Lady Chatterley’s Lover referring to an actual bluebird?

The lush, dark green of hyacinths was a sea, with buds rising like pale corn, while in the riding the forget-me-nots were fluffing up, and columbines were unfolding their ink-purple ruches, and there were bits of blue bird’s eggshell under a bush.

Solution 1:

No, a blue bird is not a bluebird. A bluebird is a type of American thrush, and there are no bluebirds in Britain where the novel you cite from is set. This is a bluebird:

as is this:

Those are both bluebirds. There are no bluebirds in Europe, only in North America.

The famous color “robin’s egg blue” refers to the American robin, not to the European one, whose eggs are the wrong color.

Furthermore, the robins of Europe are not thrushes, and do not have the blue eggs characteristic of the American robin. Rather, they have eggs that look like this:

The phrase

bits of blue bird’s eggshell

is applying blue to bird’s eggshell. So it is the blue eggshell of some bird.

It is possible that it is shell of the Common Blackbird, a European turdid and thrush whose eggs look like this:

The so-called “Common Blackbird” acts just like an American Robin, whose genus it shares. It rather looks like its cousin, too. It just has a solid brownish black breast, not the redbreast that our own Robin is so famous for. (Which is in fact orange, but why we are calling orange red—well, that’s another tale.)

Solution 2:

It seems more likely that shells from robins' eggs, which are blue, are being referred to.

Edit: (Besides striking out robins' from the above sentence in light of informative comments, I've deleted an irrelevant paragraph about robin egg blue and a robin's eggs photo, in favor of the following:)

Several species of birds are possible candidates, including the Song Thrush, Turdus philomelos. Here's a picture of their eggs and a brief description of the bird, from wikipedia:

The Song Thrush (Turdus philomelos) is a thrush that breeds across much of Eurasia. It is also known in English dialects as throstle or mavis. It has brown upperparts and black-spotted cream or buff underparts and has three recognised subspecies. Its distinctive song, which has repeated musical phrases, has frequently been referred to in poetry.