Clint Squint

"You know, if you did have a squint, it might actually improve your appearance."

—Spike, Press Gang

Every once in a while actors think that they have to scrunch up their eyes while being on screen. Reasons for that are manifold, though it most commonly seems to be a way of increasing the badass level of the character they are playing. Squeezing your eyes together works like a focusing mechanism for the Death Glare. However, some actors seem to think of the Clint Squint as some sort of cheap way to add depth to a character (which is, of course, destined to fail).

In animation, where facial expressions are often either very broad or completely absent, a stoic character's choice of expressions can be basically limited to this (as everything else is either too emotionally open, or too subtle to convey).

Other reasons for squinting include:

Filming a movie or tv-series in really bright surroundings, like deserts
Large Hams who like exaggerated facial expressions.

The trope is named for Clint Eastwood.

Examples of Clint Squint include:

Anime and Manga

Admiral Akainu from One Piece.
Seraphim from Kore wa Zombie Desu ka? is a female example.
Another female example is Saeko Busujima from Highschool of the Dead. Bonus points for being more Clint Eastwood-like (at least in personality) than the rest of her party (particularly as he appears in The Good, the Bad and the Ugly and almost appeared in Batman).

Comics

Alex Ross tends to draw and paint Superman with a Clint Squint. He also had one in the old 1940's Max and Dave Fleischer also tended to give him a Clint Squint in the 1940's Superman Theatrical Cartoons, and his DCAU appearance tends to be drawn like this in tribute to the Fleischer cartoons.
A major part of Rob Liefeld's art. So major, in fact, that it's surprising to see characters who actually have eyes.
In one Garfield comic strip, Garfield is taken aback when John seems to have gotten something in his eyes. "I'm squinting! All macho guys squint!" He then promptly walks into a doorframe in the next panel.
Lampshaded in Doonesbury: upon meeting Clint Eastwood back when he was running for mayor of Carmel, Ca., Lacey Davenport tells him "Don't squint so much, dear, you'll end up with goose feet."
Vasco, the main character in Javier de Isusi's comic book The Pipe of Marcos, does display different kinds of expressions, but a large majority are squints. Come to think of it, he's hardly the only one in that comic to do that.

Films -- Live Action

Trope Namer is Clint Eastwood, who frequently squinted while playing his iconic western roles. Eastwood said he only did it because the cigars his character smoked bugged his eyes, and he was allergic to horses.
Parodied in Back to The Future Part III where Marty McFly uses Clint Eastwood's name as well as his squint.
Lee Van Cleef often wore a perpetual sneer when playing movie villains, which involved a steely-eyed squint.
Charles Bronson, perhaps aided by his naturally narrow eyes due to Lithuanian ancestry.
Italian actor Bud Spencer always did this. He had sight problems that made it hard for him to get things in focus even at short range when not squinting. These days he has recovered some sight through surgery, and squints a lot less.
Burn After Reading: Brad Pitt's character Chad trys to give Osbourne Cox one when they first meet. He fails hilariously.

Live Action TV

Captain Archer of Star Trek Enterprise fame did this to an extent that made you want to write "insert coins here" on his (barely recognizable) eyelids.
One episode of Mystery Science Theater 3000 had Observer doing this while saying "Chilly Squint!" threateningly.
The titular Walker, Texas Ranger would get one at least Once an Episode—usually complete with dramatic close up and Theme Music Power-Up.
Parodied by George W. Bush in Late Night With Conan O'Brien with the "Live Via Satellite" skits.

Squint, please (Bush squints). Bring it on! (theme from The Good, the Bad and the Ugly plays)

Video Games

During Samara's loyalty mission in Mass Effect 2, Shepard can use the Clint Squint to intimidate a Krogan. It works rather well.
- Likewise, when we finally get to meet Admiral Hackett face-to-face, this seems to be the default state of his eyes.

Web Comics

This El Goonish Shive comic. Bonus Points for being Lampshaded in the commentary.

Western Animation

In Batman: The Animated Series, Batman is extremely stoic and hard to surprise. When in costume, nearly his only way to react to anything is narrowing his eyes.
- Notably, in an episode of Justice League Unlimited Amanda Waller calls him "Rich-boy", implying that she knows his identity, and the only response he makes is to widen his eyes.
  - And of course, the very first time you see him in the Animated series, his very first action, is to squint(0:36).
Samurai Jack does this. ALL the time.
Vendetta in Making Fiends often squints her eyes to emphasize what she's saying.