Continuous-variable squeezed light is a key resource for quantum precision measurement, quantum information processing, and quantum radar. Traditional methods of generating squeezed light, such as parametric down-conversion and four-wave mixing, often rely on nonlinear optical processes with limited bandwidth and strict operating conditions. In this work, we propose a new mechanism for generating broadband squeezed light based on the magnetostrictive effect in a cavity opto-magnomechanical system. The aim is to achieve stable, wideband squeezed output fields that are robust against environmental thermal noise, using experimentally feasible parameters. The proposed system consists of a YIG (yttrium iron garnet) microbridge embedded in a microwave cavity, with a high-reflection mirror attached to the YIG crystal to form an optical cavity. The magnon mode in the YIG crystal couples to the microwave cavity mode via magnetic dipole interaction, and to the mechanical vibrational mode (phonon) via magnetostrictive interaction. The phonon mode further couples to the optical cavity mode via radiation pressure. A strong microwave drive is applied to the magnon mode, and a red-detuned laser drives the optical cavity to activate anti-Stokes scattering, establishing a state-swap (beam-splitter) interaction between the mechanical mode and the optical mode. Using the standard quantum Langevin equations and linearization around large steady-state amplitudes, we derive the fluctuation dynamics of the system. The noise spectral density of the output optical field is calculated via the input-output formalism, and the squeezing level is evaluated as a function of detuning, phase, cavity decay rate, and environmental temperature. Numerical simulations using experimentally realistic parameters (e.g., magnon frequency ~10 GHz, optical cavity decay rate ~10 MHz, mechanical frequency ~10 MHz, and coupling strengths on the order of MHz) show that a stable squeezed output field with a bandwidth exceeding 30 MHz can be generated. The minimum noise spectral density reaches 0.16 (corresponding to 4.95 dB below the shot-noise level) under optimal conditions. The squeezing is observed over a wide frequency range around the mechanical resonance. The optimal phase angle for homodyne detection is found to be near π/2, where the strongest squeezing occurs. The influence of the optical cavity decay rate is also investigated: as the decay rate increases to approximately 30 MHz, the output field becomes stable and broadband. Notably, the squeezed light exhibits strong robustness against thermal noise. When the environmental temperature rises from 20 mK to 1 K, the noise spectral density increases only slightly from 0.16 to 0.19 (i.e., from 4.95 dB to 4.2 dB below shot noise). The best robustness is observed near 20 mK. We have demonstrated a feasible and robust scheme for generating broadband squeezed light in a cavity opto-magnomechanical system via the magnetostrictive effect. The squeezing is transferred from the squeezed mechanical mode to the optical output field through state-swap interaction. The system operates with experimentally achievable parameters and tolerates relatively high thermal noise, making it suitable for practical quantum technologies. This work provides a new pathway for integrated quantum light sources with potential applications in quantum wireless networks, quantum radar, and high-precision measurements.