Why is my bird making squeaky sounds after hatching chicks?

Thanks for your question about Blue, and thanks for the pictures. She is a beautiful bird. From your description, and from the photos, I would be concerned that Blue isn't feeling well. She looks to be thin in the photos - the way we look at body condition in a bird is by the breast, and I can see that her keel bone appears to be more prominent in the photos. This, along with the abnormal vocalizations that you are hearing, may indicate that she is sick. I would ensure that her environment is free of stress due to her having her chicks, as an increase in stress can lead to illness. If she is stressed, she may also not eat as much food as she needs as well. I would recommend that she is evaluated by a veterinarian, if possible - depending on what they find on examination, they may recommend diagnostic testing such as radiographs or systemic blood work. They can also offer potential treatment options for her as well.
To decrease stress at home, please ensure that she is in a quiet and warm place with no drafts, loud noises, or unfamiliar objects/people/animals. Ensure she has places to hide and plenty of fresh food and water within easy reach as well.
I hope that gives you a place to start, and that Blue feels better soon!