Use the US online check-in to check in for the first segment. IME, it will only let you check in for that segment, too

At the airport, possibly try speaking to a GA at US (or, better if possible, UA) to see if they can issue the rest. IME, the US kiosks CAN check you in for the whole trip and will print your UA BPs on US cardstock or paper, but Mr. Chicken cannot.