We obtain the solution of models of self-avoiding walks with attractive interactions on Husimi lattices built with squares. Two attractive interactions are considered: between monomers on first-neighbour sites and not consecutive along a walk and between bonds located on opposite edges of elementary squares. For coordination numbers q > 4, two phases, one polymerized, the other non-polymerized, are present in the phase diagram. For small values of the attractive interaction the transition between those phases is continuous, but for higher values a first-order transition is found. Both regimes are separated by a tricritical point. For q = 4 a richer phase diagram is found, with an additional (dense) polymerized phase, which is stable for sufficiently strong interactions between bonds. The phase diagram of the model in the three-dimensional parameter space displays surfaces of continuous and discontinuous phase transitions and lines of tricritical points, critical endpoints and triple points.