The fact that she immediately crawls into his arms and hugs him tells you that even at all of six years old, she knows exactly what he really means. After all, if you recall his first family scene, you know they think he's a terrible liar.
The look on Hiller's face when he finds out that the air base where his fiancee and stepson were headed has been destroyed. They're alive, but he doesn't know that, and they have a similar reaction when they think he's been killed.
The novel features an extra scene just before the counterattack briefing. Hiller tries to call Jasmine and his parents. All he keeps getting is "All circuits are busy." He then starts to blame himself for not taking Jasmine with him, leading to his determination to make the aliens pay.
The scene where Russell's teenage son overhears his dad's last message... and not being able to say goodbye after all those times of thinking his dad was nothing but a loser.
While Russell's kamikaze piloting is a truly awesome moment, the scene when he makes the decision to pilot his plane directly into the alien's energy weapon is still heartbreaking:
Russell: Do me a favor. Tell my children...I love them very much...
What makes it worse is that Russell says this after looking at a picture of his children that he's taped to the inside of his plane.
A meta example - this film was the swan song for an era. It was pretty much the last big-budget special-effects movie to rely almost entirely on practical effects and models, rather than CGI.
The viral marketing website for Independence Day: Resurgence, Warof1996.com has a map of all the cities that were destroyed. Looking through it reveals how widespread and total the devastation was. Cities like Tokyo, Paris, London, Berlin, and New York were destroyed in equal measure alongside places like Pyongyang, Riyadh, and Guadalajara. All in all, 108 cities were wiped off the map in just 48 hours, with hundreds of millions, likely billions of deaths.
Plus, once you consider all that mankind lost on top of the human dead. Mankind has also lost its countless treasures, artworks, monuments, and antiquities. The British Museum, the Smithsonian, and the Louvre were destroyed in the first wave, taking with them the works of humanity's greatest and often most obscure artisans as well as our historical and cultural heritage. Institutions that lasted for decades, even centuries, like Harvard, MIT, Oxford, etc are all gone.
Combined with Fridge Horror, in the special edition, Russell tells his ill stepson that his "mother didn't want to take her medicine, either." No further information on his wife's death is given, but it leaves open a lot of unpleasant possibilities.
While fodder for quite a bit of comedy, the scene where David is in a drunken rage and tearing a break room apart can be rough to think about. Here is a Wide-Eyed Idealist who is utterly beaten down by the universe and can do nothing in response but knock over a break room trash can, in hopes of making the world a tiny bit worse out of spite. It's only a Eureka Moment thanks to a word of concern from his father that brings him back.