It would appear that your daughter might have more than one problem, not necessarily related.
The elevated Thyroglobulin Antibodies (TgAb) is the only thyroid related test and it does indicate Hashmioto's. Did your doctor, by any chance, order Thyroid Peroxidase Antibodies, as well? Those are the other antibodies that are often (though not always) present with Hashimotos'.
Does your daughter have symptoms of a thyroid condition? Although some people do have hives with thyroid condition, the most common symptoms are those of hypo or hyperthyroidism. For hypothyroidism, symptoms include fatigue, weight gain/inability to lose, swelling/puffiness, especially around the eyes, muscle/joint aches/pains, cold intolerance, constipation, etc. For hyperthyroidism, common symptoms include fatigue, heat intolerance, rapid weight loss, irritability, mood swings. diarrhea, hand tremors, etc.
Depression, anxiety, rapid heart rate or heart palpitations can apply to either, but are most often considered to be symptoms of hyper.
BUN and BUN/creatinine ratio is related to blood flow to the kidneys/kidney function. BUN can be elevated if the patient is dehydrated when the blood is drawn, such as when one has been fasting overnight. Many people don't realize that, unless instructed otherwise, it's, typically, okay to drink plain water prior to a fasting lab draw in order to prevent dehydration. Although elevated, the BUN is not "very" high, so the doctor may ask for a repeat of this test or do further tests to make sure there is no kidney malfunction.
EOS is a type of white blood cell that fights disease. Its presence can indicate a parasitic infection, an allergic reaction or a cancer. High EOS can be present in blood or at the site of infection or inflammation. High EOS could be present because of an autoimmune condition, such as Hashimoto's or they could be elevated because of inflammation caused by an allergy/infection, etc.
Immunoglobulin E (IgE) is a type of antibody.
"Elevated concentrations of IgE are generally thought of in the context of allergic disease. However, increases in the amount of circulating IgE can also be found in various other diseases, including primary immunodeficiencies, infections, inflammatory diseases, and malignancies. Total IgE measurements have limited utility for diagnostic evaluation of patients with suspected allergic disease, except for allergic bronchopulmonary aspergillosis (ABPA). ABPA is a hypersensitivity reaction against the fungi Aspergillus that occurs most frequently in patients with asthma or cystic fibrosis. An elevation of total IgE is part of the diagnostic criteria for ABPA, although the specific diagnostic concentration is dependent on certain patient characteristics."
"Elevated thyroid antibodies are often associated with chronic urticaria, also called hives. Studies report that as many as 57.4% of patients with hives have the presence of anti-thyroid antibodies. An August 2010 paper suggests that treatment with T4 improves the itching associated with urticaria, but did not advise treatment with T4 unless the patient was hypothyroid." - excerpt from Life Extention - Thyroid Regulation
My hives didn't itch either actually.