JOHN YANG: Israel's intensified assault on Gaza comes even as an estimated 600 American citizens are still trying to leave.

They want to use the Rafah Border Crossing in southern Gaza to enter Egypt.

Well, Egypt and Hamas are allowing trucks delivering aid to Gaza to use the crossing.

They're still refusing to let foreign nationals use it to leave.

Tonight, Nick Schifrin brings us the story of an American mother and her three small children who are trapped in Gaza and the husband and father who anxiously awaits them at home.

NICK SCHIFRIN: The Okal family was never supposed to be here.

Eight year old Iyad, two month old Elias and two year old sister Nadine.

They were never supposed to be on the move again running from Israeli airstrikes.

Their mother 32-year-old Haneen helps load the car to move homes for the fourth time in three weeks.

We filmed this on Thursday, before Gaza's internet was cut.

HANEEN OKAL, American-Palestinian Stuck in Gaza: It's so stressful.

My kids are so scared.

Very terrified.

My son didn't sleep at all last night.

They were bombing all over near our house and it's so scary in there.

I don't know where to go.

NICK SCHIFRIN: In the car, Iyad looks into the camera.

His eyes wander but he cannot escape this crisis.

The stare of a boy who has seen too much, who's been forced to grow up too fast.

The airstrikes chased them from their family home in northern Gaza and then Gaza City, or some neighborhoods are now Moonscapes.

An IDF warning drove them further south to Khan Yunis and then Rafah, Gaza's border with Egypt.

In the car, Haneen call comforts her baby.

But there's no comfort in children who are old enough to understand.

IYAD OKAL: It's sad.

And I just don't know who to describe it.

It's like all damaged and destroyed and all bombed.

God may help us.

The American citizens can get it as soon as possible.

And we can go to Rafah border.

And that's my wish.

That's my wish to Biden.

NICK SCHIFRIN: It wasn't that long ago that Iyad's wish was to play better Mozart, or share how his heart will go on with his 86,000 followers on Instagram.

Or just spend the day with his dad, Abdulla.

The family lives 5,500 miles away in New Jersey, other than the youngest, they were all born in the US.

IYAD OKAL: I feel sad that I can't see my dad.

I miss him so much.

He missed me.

ABDULLA OKAL: It's very hard.

When they tell you when your kids keep on telling you come get us, come get us and you can do anything to them.

I feel like broke, broken.

NICK SCHIFRIN: We spoke to Abdullah on Friday night here in the U.S. as Israeli ground forces increased incursions into Gaza.

ABDULLA OKAL: When they go on the ground it will be very ugly.

It will be a bloodbath.

So now it makes me worried more worried, you know, more scared.

There's no food, no water, no electricity.

I don't know how long they're going to survive.

If they don't die from a bomb, they will die starving.

NICK SCHIFRIN: When your son asks you to meet him to take him out.

What do you tell him?

ABDULLA OKAL: I tried to make up stories.

I don't mean no -- I'm trying to get an airline ticket for you guys.

There is no airline tickets.

Basically, I'm lying to him.

I can't.

I don't want to tell him.

Hey, you're an American citizen and our government cannot evacuate them.

But they can send bombs.

NICK SCHIFRIN: At first Western officials say Egyptian President Abdel Fattah El-Sisi resisted opening Rafah.

ABDEL FATTAH EL-SISI, Egyptian President: I need to stress the importance of not allowing Palestinians to be displaced from Gaza into Sinai.

NICK SCHIFRIN: But now U.S. officials insist it is Hamas, who's preventing Americans from leaving Gaza.

The U.S. government says it's Hamas that is blocking people like your family from getting through to Egypt.

ABDULLA OKAL: No one stopped them from going to the crossing border.

So I need explanation.

I need to know if they say in Hamas.

When I go ask my wife and ask others, they say no, nobody's stopping them.

I spoke with the State Department they call every other day to get the updates from me.

I am - - I supposed to get the update from them.

NICK SCHIFRIN: These are the most recent videos that Abdulla has of his family.

He hasn't been able to reach his wife since Friday morning.

ADBULLA OKAL: And she's hopeless.

She thinks that we're not going to meet again.

I wish if the border open, I'm willing to go and stay with them.

At least either we live together or we die together.

But this is what we've gone through.

NICK SCHIFRIN: Abdulla calls his family homeless, but it is he too who feels homeless without them.

For the PBS News Weekend, I'm Nick Schifrin.