In a cell-free cloud radio access network (C-RAN) architecture, uplink channel estimation is carried out by a centralized baseband processing unit (BBU) connected to distributed remote radio heads (RRHs). When the RRHs have multiple antennas and limited radio front-end resources, the design of uplink channel estimation is faced with the challenges posed by reduced radio frequency (RF) chains and one-bit analog-to-digital converters (ADCs) at the RRHs. This work tackles the problem of jointly optimizing the pilot sequences and the pre-RF chains analog combiners with the goal of minimizing the sum of mean squared errors (MSEs) of the estimated channel vectors at the BBU. The problem formulation models the impact of the ADC operation by leveraging Bussgang's theorem. An efficient solution is developed by means of an iterative alternating optimization algorithm. Numerical results validate the advantages of the proposed joint design compared to baseline schemes that randomly choose either pilots or analog combiners.