Did you strip the firing pin out of the bolt and clean the firing pin passage?
If not, start there. Don't be shy, either. Get a bunch of toothpicks, or some copper wire, and dig around in there, looking for compressed greasy mung that has been pounded nearly solid.
Got a digital camera? Take a couple pictures. Easy reference material for re-assembly.
JC Higgins was Sears house brand, They did not make any guns. Find the model number, find out who actually made it and what model it is, and find the parts diagram for it online. Google images search is pretty good at finding what you need.
Still too nervous to take the trigger apart? Get a nice new can of WD-40, put the tube on the nozzle, take it somewhere other than in your home, and hose the crap out of the trigger and all the area, with the WD-40. Be generous! After the WD treatment, find an air hose with a nozzle on it, and blow out the trigger area. Then lube with gun oil.
WD-40 is mostly solvents, and is designed as a Water Displacer (WD!). It has a little bit of oil in it, but if you trust it to keep your guns from rusting, you get what you get.
If you don't want or have access to an air nozzle, look at getting a can of brake cleaner. Pure solvents, bad for you, and flammable. Not for indoors! Oil after all is dry again!
Most 22's can stand a good cleanout on a regular basis. The ammo tends to be pretty dirty, and the residue builds up.
Apartment dweller? A cardboard box with a couple old t-shirts or towels inside makes an OK place to spray out this kind of project, out on the deck.
Wash out the innards, if the problem continues, you may be stuck doing a detailed strip and clean. Usually, this would be a good time to be looking for worn or damaged parts.
May wish to try some other ammo too. Some folks have had issues that went away with an ammo change. I have not, but it may be worth a try.
For the most part, if you go carefully and pay attention to what you are doing, there are not too many things you can do to the innards of a 22 that will cause you a bunch of issues.
Once you track down the maker and model number, you can get a better idea of what to look for. There is a paperback book available that details several common models. Usually in stock at any gun store that has a book shelf. Need to know the make and model though.
This is the book I think may be of most help.
http://www.books-by-isbn.com/0-8734...Firearms-Assembly-Disassembly-Part-3-Rim.html I really can't explain the price. Wow! There are some rip-off artists in the used book trade, apparently!
As books go, it's worth the $20 it should cost. Otherwise, all the info you should need, should be online somewhere.
Cheers
Trev