Okay, I'm totally editing this answer. My first answer was basically: products like 3-in-1 and silicone spray are great for loosening up stuck parts, but too light for long-term lubrication, pack it with grease (lithium grease, axle grease, pick your poison). I still stand by that...
However; what if you went high-tech and installed a couple of actual bearings? :-)
The picture is a couple of high tech racing bearings that go on something like a Porsche 911. They just looked like something that would be easy to clamp onto the swing, with small enough journals to run a bolt through and clamp them up nice and tight. There are lots of other bearings to choose from. You could even find some teflon bearings or even teflon tubes just the right size for the bolts--no lubrication ever again.
Something like 3-in-1 oil or even silicone lubricant spray is great for penetrating rust and getting stuck parts to start moving again, but long-term I think you might want something a little heavier. I have doubts about the powdered graphite.
Motor oil is actually one of the best lubricants for car door hinges, for whatever that's worth.
Personally, I'd probably clean those up with either a solvent (which will evaporate) or motor oil, or both, then pack them lightly with axle grease. I'd do it just like you were packing wheel bearings, so that you get it squeezing out the other side, then wipe away the excess on the outside. I just think that would last longer than just about anything else, and protect the metal from rusting, which is where a lot of the squeak is coming from.
Lithium grease might be really good, too. It or axle grease, same difference probably.
As for silicone spray (garage door chain lubricant), I've used it on my motorcycle chain and it's okay-ish. It's convenient because I can put a newspaper or rag behind the chain and spray it in there to clean and lubricate without getting it on everything else. But it doesn't seem to really last out in the elements. I'm going to stop that and go back to a regimen of putting the bike up on a jackstand, cleaning the chain with a soapy brush, rinsing and drying and then soaking it in engine oil.