The growing scale of genomic data poses significant challenges in terms of storage, compression, and real-time analysis. Traditional digital systems struggle to meet the demands of ultra-efficient, low-energy biological data processing. In this paper, we introduce BioQuantumMesh, a novel hybrid biocomputing system that integrates DNA-based logic circuits with quantuminspired reversible computation. This framework leverages the parallelism of molecular biology and the reversibility of quantum gates to enable high-speed, low-loss genomic data compression and pattern recognition. Our architecture simulates quantum behavior using synthetic DNA logic gates and implements adaptive data mesh networks inspired by synaptic communication in neural tissues. We further explore potential applications in realtime cancer genome analysis, virus surveillance, and large-scale bioinformatics.