Should all my home's GFCI outlets be connected to a GFCI breaker?
No.

You should not put a GFCI outlet on a GCFI breaker.  You need GFCI protection in only one device per circuit.  Or stated better, you should not have a GFCI outlet downstream of a GFCI breaker.  It's not unsafe per se, but it will be an unpredictable pain to deal with when it trips and you don't know where to go to reset it.

GFCI Outlets do not require GFCI breakers.  GFCI Breakers make regular outlets and other devices GFCI protected.