You might also guard with only one hand when:
you're wanting to use your other hand for other purposes no directly involved with fighting: getting out identification or money or whatever might help de-escalate a situation, removing some kind of restraint, taking of a belt to use as a deterrent / weapon, restricting a drunk aggressive friend from stepping into the fight etc., getting out a phone to call the police, take a photo/video as evidence etc.
you have two opponents on different sides of you, and you effectively take on a one-handed guard against each (this is a passive thing to do; if you go on the offensive, you'd shift towards one opponent and use both arms to overpower them whilst staying out of reach of the other opponent)
I wouldn't consider punching a secondary purpose: a skillful person can hit extremely hard with a jab, and it doesn't have to be an "eye jab" - using your fingers has more risk of injury to yourself and is easier for them to flinch away from, and the arm tends to be extended then the fingers snapped forward at the wrist (much less likely to break your fingers than trying to keep the fingers in line with the forearm and rigid throughout) so the arm stays out a little longer than for a punch leaving it more vulnerable to being grabbed or a counter underneath to the ribs; a closed-fist punch towards the nose is a better choice in my opinion; and don't forget the front hand can deliver a very powerful ridge-hand strike, an elbow (if you're closing distance - especially useful if they block near your wrist as you close), backfist, uppercut; you can also e.g. grab someone's under the elbow or the back of their neck, and pull yourselves together for a knee.
Of course, being able to block effectively is equally important.
If you're using just one hand, you'll find it much more effective if you're able to burst forwards and close the gap unexpectedly; then if your timing is good you should get a free hit in on anyone trying to get close to hit you. A good jab should knock them down. To get that kind of explosiveness when the front hand is reaching, you've got to have your stance just right. I looked through hundreds of google image matches on fighting stances and didn't find anything that was both good and not cropped hiding the feet), but basically you want a half-side-on back stance, where the back foot is tucked under you a bit more than usual with the foot flat and facing almost forwards, knee bent and weight settled down but not excessively, so you can get a thrusting feeling by straightening the back leg - much like pushing off the starter block for a sprint - that will launch you explosively forwards. The front leg should be bent too. As you jab, you'll snap the hips from that half-frontal position to a more side-on position, helping you cover distance and increase reach. So, it's a fairly short and narrow stance that should expand and "unwind" as the hips rotate, giving you good, explosive reach. Body weight distribution starts around 60/40 back/front, but the back foot is aligned more front-to-back, so it can take more weight towards the ball without steeping first; it's very important to be light enough on the front foot to pick it up to teep/front-kick/side-kick or jam an incoming kick.
If you have multiple opponents, you clearly can't "aim" such a back stance so you can burst forwards towards any of them, so you're either orienting towards the dominant or closest opponent, or adopt a lighter more upright stance - more like normal standing - in which case your jab will be more telegraphed and have much less explosive reach, but you'll be able to move more freely in any direction.
If someone doesn't have conditioned knuckles, a palm strike isn't a bad idea, but then you've really only got the chin and nose as targets. If you're trying to delay the fighting whilst you get something with your other hand, it's psychologically less aggressive to raise an open hand though, compared to a closed fist....
"Static or twitchy" is orthogonal to the use of one hand - do whatever you've trained to do.