You can test for gapeworm.. it is very rare. How old are the chickens?
Lots depends on their age... how old are they? Have you ever wormed them? When? With what? Did you repeat?
Search here on BYC for "the Q-tip test"... or "testing for gapeworm".. they rarely have it. And, the one picture you posted looks like a young bird.. ?
You can search "worming with Valbazen"... "worming"... "giving oral medications".. "dosages of Valbazen".. "how much Valbazen"... "how to give food and water orally".
You can get Valbazen at the local feed store.. it is a bit expensive.. also pick up some syringes (no needles)... also available at the feed store... note info posted by dawg53 on worming when you search. Antibiotics are also available at the feed store -- the Duramycin 10 (Tetracycline) is water soluable, goes into the drinking water.. for several days... is also available at the feed store.. they told me 1 tsp per gallon..... in only the amount of water they will drink in 24 hours.. so.. just one gallon or half a gallon, depending on your birds.. but, if they won't drink it... you may have to give it to each one orally as well. Be careful to avoid the windpipe.. if you get water into the airway (or food, or medicine, or anything).. they will die rather quickly.
Tylan 50... with needles and syringes.. you can give it in the breast muscle.. (.5 cc/ml -- 1/2 cc/ml per standard chicken... less for bantams, half as much) - alternating sides each day.. for three days.. stop for three and if needed give it again for three more... but, if they have worms.. antibiotics will only perk them up for a little while.
You can search "giving injections" and "Tylan 50" here also.
If they are young... five months or younger... then, it's probably not worms and you can search the other symptoms... "respiratory symptoms", "wheezing"....
Keep an eye on their poop as well... and search here "normal poop".. there is a site with pictures you will find the link to. If they are old enough to have worms.. maybe start water antibiotics and worm them.. you have to repeat worming in ten days and withhold eggs.. more info in search. Again, see dawg53 posts on worming.
Searching "lethargic" won't get you much info... or it gets you too much, as it is too common... too many causes... it's not specific enough.
If they won't eat or drink... you can lose them quickly... Good to act asap to save them.... Feed stores generally open early .. about 8am... call to see in the am.
Good luck...