The Fourier Neural Operator (FNO) has attracted significant attention for solving partial differential equations in solid mechanics due to its strong generalization capability across varying inputs. However, its reliance on the Fast Fourier Transform (FFT) restricts FNO to regular computational domains, which severely limits its applicability to mechanical problems involving complex geometries. This paper develops a Finite-Cell-based Physics-Informed Fourier Neural Operator (FC-PINO) framework to overcome this limitation. The key idea is to integrate the Finite Cell Method (FCM) into the FNO architecture. In this approach, irregular physical domains are embedded into a regular, fictitious domain, enabling the use of structured grids that are fully compatible with the application of FNO while accurately representing complex geometries. The paper focuses on equilibrium problems of hyperelastic materials, for which a potential energy functional exists and equilibrium configurations correspond to minimizers of the total potential energy. Within the machine learning framework, the physical domain is discretized using FCM into finite cells, which serve as the computational grid for the neural operator. Cell partitioning and adaptive integration techniques are employed to accurately evaluate the strain energy and to capture geometric boundary effects introduced by complex domain shapes. The total potential energy of the system is adopted as the loss function, enabling physics-informed training without the need for labeled solution data. Numerical examples involving hyperelastic structures with complex geometries demonstrate that the proposed FC-PINO method can efficiently and accurately predict nonlinear structural deformations, highlighting its effectiveness and flexibility for geometry-complex solid mechanics problems.