Forcecage stops any area of effects from entering or exiting it
PHB p. 204 explains how to determine if something is considered part of an AoE or not, under the "Areas of Effect" heading (or here in the basic rules):
If no unblocked straight line extends from the point of origin to a
location within the area of effect, that location isn’t included in
the spell’s area. To block one of these imaginary lines, an
obstruction must provide total cover.
And the box of force counts as a source of full cover, per the rules for Cover (PHB p. 196, or here in the basic rules):
A target has total cover if it is completely concealed by an obstacle.
The area inside the box, being completely surrounded by the box, effectively has full cover against all spells and spell effects no matter if they are targeted or AOE. No "unblocked straight line" can possibly get from the outside of a box to the inside. It is physically obstructed from all sides.
The only exceptions to this are spells which ignore cover completely, including total cover, such as sacred flame.