Ultrasound in point-of-care lung assessment is becoming increasingly relevant. This is further reinforced in the context of the COVID-19 pandemic, where rapid decisions on the lung state must be made for staging and monitoring purposes. The lung structural changes due to severe COVID-19 modify the way ultrasound propagates in the parenchyma. This is reflected by changes in the appearance of the lung ultrasound images. In abnormal lungs, vertical artifacts known as B-lines appear and can evolve into white lung patterns in the more severe cases. Currently, these artifacts are assessed by trained physicians, and the diagnosis is qualitative and operator dependent. In this article, an automatic segmentation method using a convolutional neural network is proposed to automatically stage the progression of the disease. 1863 B-mode images from 203 videos obtained from 14 asymptomatic individual,14 confirmed COVID-19 cases, and 4 suspected COVID-19 cases were used. Signs of lung damage, such as the presence and extent of B-lines and white lung areas, are manually segmented and scored from zero to three (most severe). These manually scored images are considered as ground truth. Different test-training strategies are evaluated in this study. The results shed light on the efficient approaches and common challenges associated with automatic segmentation methods.