No, Freedom of Movement does not counteract a Forcecage
Freedom of Movement has five principle effects:
- Target is unaffected by Difficult Terrain
- Target cannot have their speed reduced [by magical effects]
- Target cannot be Paralyzed or Restrained [by magical effects]
- Target can automagically escape from non-magical bindings using 5' of movement
- Target does not get disadvantage from underwater combat
None of these effects are consequences of the effects of Forcecage, which only creates a [magical] barrier that the creature cannot pass through. As a result, Freedom of Movement has no effect on a creature trapped by a Forcecage.
Freedom of Movement doesn't give a creature carte blanche to move wherever it wants (e.g. through walls), it just prevents some specific effects (in addition to allowing escape from grapples and non-magical restraints for a cost of 5ft of movement):
- extra movement costs from difficult terrain
- reduction of speed caused by spells or magical effects
- the paralyzed condition caused by spells or magical effects
- the restrained condition caused by spells or magical effects
- penalties to movement or attack due to being underwater
A Forcecage does not reduce the speed of creatures inside it, and creatures inside of it are neither paralyzed nor restrained. Essentially, it's just a collection of six walls of force which happen to be arranged in such a way that it's really inconvenient for anyone in a 10- or 20-foot cube.