Modern embedded systems face increasingly sophisticated attacks targeting boot chain vulnerabilities, necessitating rigorous quantitative security evaluation frameworks. Existing approaches lack comprehensive metrics, standardized evaluation methodologies, and hardware-accelerated collection mechanisms with formal completeness guaranties. We present the Security-Quality Scorecard (SQS), a novel framework that integrates six complementary security metrics with formal verification using Z3 SMT solver for holistic evaluation of boot chains. Our framework introduces: (1) mathematically rigorous Formal Coverage Score (FCS) quantifying test completeness via satisfiability modulo theories, (2) hardware/software co-design for real-time metric collection via custom RTL modules, and (3) probabilistic security testing across normal operation and attack scenarios. Through extensive hardware measurement on Zynq-7000 SoC across 75,000 test samples (three independent replications, 25,000 each) covering 10 security situations, we demonstrate 96% formal coverage score with 100% state coverage (11/11 states) and 100% transition coverage (33/33 transitions). Probabilistic attack simulation achieves a Boot Integrity Score (BIS) of 42.66% ± 0.67% (CV=1.58%), validating both functional correctness (10,666 ± 168 successes) and security resilience (zero bypasses in attack scenarios). Three independent hardware replications demonstrate reproducible measurement with natural statistical variation, confirming the reliability of the framework. The hardware acceleration enables cycle-accurate metric collection with minimal overhead. Our implementation facilitates reproducible security evaluation and standardization across heterogeneous embedded platforms.