Rule of Cool is always worth considering
When it comes to the rule of cool approach, there's really no harm in deciding to interpret a critical roll as something really cool. This could be anything from giving some sort of bonus discovery when doing a perception/investigation check to search a location, to managing to knock out a mook when trying to grapple them.
Letting the dice inspire the cool moments can help to enhance the feeling that there's more to what happens than mere PC decision - luck plays a role in events, too. This doesn't mean that every nat 20 must become a cool moment, but that a nat 20 should be an additional prompt to ask "can this moment be made more cool?"
This is the way that one of my DMs likes to run it. Natural 20s get rewarded if he can see a cool bonus effect quickly. If you're rolling to remember a mundane piece of information, it's not likely to get a rule of cool bonus... but if you're trying to knock down a door with athletics, you might manage to do it at just the right time to knock an enemy over behind the door.
It's important that this doesn't become a requirement for "rule of cool" being invoked - it should only ever be a catalyst, something that reminds you to consider the rule of cool.
Rule of Uncool must be very carefully used
When people play D&D, they generally want things to feel cool. Critical fails can sometimes contribute to it feeling cool. But if every natural 1 means that you hurt yourself or an ally, for example, it will become tiresome very quickly, for most tables.
Instead, think of it as a "Rule of Funny" moment. A natural 1 on a perception check to look for enemies might see you watching a squirrel dart through the forest instead. A natural 1 on an athletics check to bust through a door might see you fall on your butt (harmlessly, but feeling quite embarrassed).
Players should never feel like they shouldn't try things because "what if I roll a natural 1?" - a 5% chance of something actively harmful or problematic isn't most people's idea of fun.
It can be quite funny, sometimes, for a rule of funny to be a rule of uncool. A player tries to demonstrate their skill with a knife by juggling, rolls a natural 1, and cuts their hand for 1d4 slashing damage... that's funny and uncool. A player tries to attack an enemy, rolls a natural 1, and breaks their weapon... that's not funny, and unless it's meant to be a feature of the weapon, or of the enemy, it feels like you're being punished for one of the fundamental parts of D&D.
As others have said, it comes down to what the table thinks falls within cool/funny. If the players (including the fighter) find the idea of a natural 1 causing the fighter to drop their weapon to be funny, then do it. If the fighter is going to feel like they're being punished for fighting, then don't... most of the time.
It can always be funny to have it happen once, in a controlled context. The Level 9 fighter tries to strike down a simple goblin after the goblin says something that offends her, rolls a nat 1, and manages to fumble the weapon, which the goblin grabs, leading to a chase to reclaim the weapon? That can be hilarious... once. If it happens on every nat 1, it's just not funny anymore. Especially when, at high levels, the fighter ends up with an 18.5% chance of it happening at least once, on EACH turn. In 1 minute of fighting, it would happen at least once more than 70% of the time.