You could use a 243 suppressor for a 223. You'd be ok shooting your 22 thru it as well, but it wouldn't be ideal, and the dirty concern above should be researched.
22LR cans are made to be taken apart and cleaned where most center fire cans are not. Another thing with 22 is most ammo is uncoated lead and causes a lot of lead fouling in the can and it needs to be cleaned occasionally.
To answer your question yes you can but the 22LR is not advised.
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