Is there an intuitive reason as to why there is no phase transition to get to a degenerate Fermi gas? Cooling a bosonic gas leads to a phase transition into the Bose-Einstein condesate. This is characterised by a symmetry broken ( U(1), by choosing a specific phase for the macroscopic wavefunction) and by discountinuous behavoiur of some thermodyamics quantities and their derivates - e.g. the heat capacities.
On the other hand, cooling a fermionic gas leads to a crossover into the a Fermi degenerate gas.
Is there an intuitive reason as to why the latter does not constitute a phase transition?
 A: In terms of phase transitions, interactions are the main source of intuition.  You can imagine a collective field which biases the order parameter.  For non-interacting bosons, the collective field is coming from the statistics instead of an interaction (an energy).  Thus, it’s hard to identify the collective field as a term in an action or free energy.  Instead we have to think more intuitively about these statistics and how a collective field forms and acts on its constituents.  First the statistics of indistinguishable particles can be understood as distinguishable particles communicating with each other by constructive and deconstructive interference.  For fermions the “signal” communicated is different from each particle as each particle is in a different state,  thus the collective interference pattern cast by the N-1 particles on the N particle is noisy and random. While for bosons the signal is the same from each particle and the interference pattern reinforces its self.  This reinforcement is the same sort of feedback process that interactions produce when forming a collective field in a classical phase transition.
For interacting systems,  the collective inter-particle interference is not the only way particles communicate. In bosinic systems the suppression of thermal excitations is further reinforced by interactions and produces a gap in the excitation spectrum.
While interactions produce no collective field in interacting fermions for completely repulsive interactions.  For attractive interactions,  fermions pair into bosons and then condense. 
