The incident happened around 7:30 p.m. on Boulevard East in North Bergen.
The children were playing in the apartment when they saw a friend playing on the ground below and climbed onto a bed to get a better view.
Authorities say they leaned against the screen, and that's when it gave way.
"I saw people screaming and running," witness Christina Saoud said. "Two kids on the ground and mother coming from the building crying and screaming."
They lay motionless and crying as their mother raced to help them.
"It was sad, really sad, like a parent's worst nightmare," witness Mya Saoud said.
Officials say there are encouraging signs the kids will pull through.
The children were both conscious at the scene, but crying hysterically. The girl was complaining about a possible broken leg.
The children were rushed to Jersey City Medical Center before being transferred to University Hospital.