This paper studies the consensus problem of high-order multi-agent systems (MASs) with binary-valued communications and switching topologies. To tackle the challenge of unknown states caused by binary-valued communications, this paper constructs an estimation-based consensus algorithm. First, a recursive projection identification algorithm is presented to estimate the neighbors' states dynamically. Then, based on these estimates, a consensus law is designed. By constructing and analyzing two combined Lyapunov functions about estimation error and state error, this paper establishes their relation, to overcome the difficulty resulting from the coupling of the estimation and control and less information due to switching topologies. Under the condition of jointly connected topologies, it is proven that by properly selecting the step coefficient, the estimates of states can converge to the true states with a convergence rate as the reciprocal of the recursion times. Besides, the MAS is proved to achieve weak consensus and the consensus rate is also established as the reciprocal of the recursion times. Finally, a simulation example is given to validate the algorithm.