This paper considers the component carrier selection (CCS) problem for an LTE-U and WiFi coexistence network operating in an unlicensed spectrum band. An unlicensed component carrier selection (U-CCS) algorithm is proposed to select component carriers (CCs) for newly arrived LTE-U users. To support the U-CCS algorithm, an implementation of the carrier sense adaptive transmission (CSAT) mechanism is first presented to coordinate the transmission of LTE-U users and WiFi users on a single CC. The proposed U-CCS algorithm takes both carrier load and channel condition into account and aims to find a CC among all CCs at the arrival of a new LTE-U user, on which the user can achieve the maximum throughput, while ensuring that the average throughput of each WiFi user on the same CC meets the minimum throughput requirement. Simulation experiments show that the proposed U-CCS algorithm outperforms a couple of existing CC selection algorithms in terms of the minimum LTE-U user throughput, minimum WiFi user throughput, and newly arrived user throughput.