The outbreak of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic highlighted the serious problems of health care systems but also threatened the mental and physical health of patients worldwide. The goal of this study was to assess psychological health and insomnia in people with chronic diseases in the time of elevated stress associated with the pandemic. The study involved 879 people from Zachodniopomorskie province in Poland. Each participant provided basic demographic data, data on symptoms of insomnia, depression, anxiety and information on concomitant diseases such as hypertension, diabetes mellitus, coronary heart disease, heart failure, dyslipidemia, chronic obstructive pulmonary disease, Hashimoto’s disease and smoking cigarettes. Chronic diseases included in this study showed a strong correlation between Hashimoto’s disease and increase scores according to the Insomnia Severity Index (ISI, r = 0.797, p < 0.001), the Generalized Anxiety Disorder scale (GAD-7, r = 0.766, p < 0.001) and the Patient Health Questionnaire (PHQ-9, r = 0.767, p < 0.001). After the results were corrected for age, gender, diagnosed hypertension, dyslipidemia and cigarette smoking, it was confirmed that the diagnosis of Hashimoto’s disease was associated with an increased risk of anxiety (odds ratio (OR) = 2.225; p < 0.001), depression (OR = 2.518; p < 0.001) and insomnia (OR = 3.530; p < 0.001). Our study showed that during the SARS-CoV-2 pandemic patients with Hashimoto’s disease show a higher risk of insomnia, anxiety and depression.