The seven children of the title live in 1880s Sydney with their father, an army Captain who has little understanding of his children, and their 20-year-old stepmother Esther, who can exert little discipline on them. Accordingly, they wreak havoc wherever possible, for example by interrupting their parents while they entertain guests and asking for some of their dinner (implying to the guests that the children's own dinner is inadequate).
After a prank by Judy and Pip embarrasses Captain Woolcot at his military barracks, he orders that ringleader Judy be sent away to boarding school in the Blue Mountains.
Meg comes under the influence of an older girl, Aldith, and tries to improve her appearance according to the fashions of the day. She and Aldith make the acquaintance of two young men, but Meg believes she has fallen in love with the older brother of one, Alan. When Aldith and Meg arrange to meet the young men for a walk, Meg is embarrassed after a note goes astray and Alan comes to the meeting instead and reproaches her for becoming 'spoilt', rather than remaining the sweet young girl she was. Meg returns home and later faints, having tight-laced her waist under pressure from Aldith until it affects her health.
Unhappy at being away from her siblings, Judy runs away from school, returns home, and hides in the barn. Despite her ill-health as a result of walking for several days to get home, the other children conceal her presence from their father, but he discovers her. He plans to send her back to school, but when the doctor reports she has pneumonia and is at risk of tuberculosis, she is allowed to remain at home.
To assist Judy's recuperation, Esther's parents invite her and the children to their sheep station Yarrahappini. One day the children go on a picnic far away from the property. A ringbarked tree falls and threatens to crush 'the General', the youngest sibling and Esther's own child. Judy, who promised 'on her life' not to allow him to be harmed on the picnic, rushes to catch him and her body protects him from the tree. However her back is broken and she dies before help can be fetched.
After burying Judy on the property, the family returns home to Sydney sobered by her death. While ostensibly things remain the same, each character is slightly changed by their experience. In particular Captain Woolcot regrets the fact that he never really understood Judy and tries to treasure his remaining children a little more, though he shows it very little more than before.

Question: Why is Meg embarrassed during a meeting?
Answer: The wrong brother, Alan shows up.
Alternative answer: Meg was embarrassed because Alan  was not approved of her beehaviour.