Control flow in unstructured programs can be complex and dynamic, which makes static analysis difficult. Yet, automated reasoning about unstructured control flow is important when certifying properties of binary (machine) code in trustworthy systems, e.g., cryptographic routines. We present a theory of forward symbolic execution for unstructured programs suitable for use in theorem provers that enables automated verification of both functional and non-functional program properties. The theory's foundation is a set of inference rules where each member corresponds to an operation in a symbolic execution engine. The rules are designed to give control over the tradeoff between the preservation of precision and introduction of overapproximation. We instantiate our theory for BIR, a previously proposed intermediate language for binary analysis. We demonstrate how symbolic executors can be constructed for BIR with common optimizations such as pruning of infeasible symbolic states. We implemented our theory in the HOL4 theorem prover using the HolBA binary analysis library, obtaining machine-checked proofs of soundness of symbolic execution for BIR. We practically evaluated two applications of our theory: verification of functional properties of RISC-V binaries and verification of execution time bounds of programs running on the ARM Cortex-M0 processor. The evaluation shows that such verification can be automated with moderate overhead on medium-sized programs.