We investigate numerically the micromagnetic properties and the low-energy physics of an artificial square spin system in which the nanomagnets are physically connected at the lattice vertices. Micromagnetic simulations reveal that the energy stored at the vertex sites strongly depends on the type of magnetic domain wall formed by the four connected nanomagnets. As a consequence, the energy gap between the vertex types can be partially modified by varying the geometrical parameters of the nanomagnets, such as their width and thickness. Based on the energy levels given by the micromagnetic simulations, we compute the thermodynamic properties of the corresponding spin models using Monte Carlo simulations. We found two regimes, both being characterized by an extensive ground state manifold, in sharp contrast with similar lattices with disconnected nanomagnets. For narrow and thin nanomagnets, low-energy spin configurations consist of independent ferromagnetic straight lines crossing the whole lattice. The ground state manifold is thus highly degenerate, although this degeneracy is subdominant. In the limit of thick and wide nanomagnets, our findings suggest that the celebrated square ice model may be fabricated experimentally from a simple square lattice of connected elements. These results show that the micromagnetic nature of artificial spin systems involves another degree of freedom that can be finely tuned to explore strongly correlated disordered magnetic states of matter.