Do MOSFETs have a diode built into them?
Yes, power MOSFETs have a parasitic diode called Body Diode.



As a result of this diode, a single MOSFET can work only as a unidirectional switch.  A single MOSFET can't switch-off the opposite direction, because the diode conducts independent of the gate.

The body diode is usually fairly slow to turn on.  I advise against using it as a the only flyback diode.  Add an external Schottky diode for flyback protection.

(There are power MOSFETs with a built-in Schottky diode.  They have 2 diodes: the native body diode, and an additional Schottky.  But such MOSFETs are relatively uncommon.)