When it's cold like this it gets tough. If you have a trapping supply place near by, you can find some lure/bait that has glycerin added to prevent freezing.... some sporting goods places carry this type of lure as well.
Maybe some fishy smelling catfood or sardines?
Often with cubbys (boxes in this case) I want the scent to flow through, rather than just staying in the cubby/box, so having some air flow across the bait helps, so a hole in both ends of the boxes helps with this, so maybe hardware screen on the bait end hole, and trap end hole left open.
For the 110s the boxes need to have a slot in the sides to allow the springs to protrude, something like this picture... note that you can build a long box with a trap on each end and bait in the center... also be sure to bend and spread the wire triggers out a bit
Typically, when targeting a specific pest critter, I'll set heavy using a lot of traps, and "chum" it a bit, by placing some nibbles of bait/meat out in front of the set... in the case of the weasel boxes maybe a nibble sitting right in the hole of the box... in the case of the 110s maybe a bit poked on the trigger wire and slid up/down.. note you can flip the 110s so the trigger is on the bottom, so the critter has to climb over too.
Sometimes I'll chum heavy for a night or two with no traps set, often with a game camera set up... this can tell you a lot about how to adjust, and eliminates the critter getting trap shy due to a miss fire, etc. Once it's coming and going with confidence you can set the traps... this is a good way to deal with multiple individuals as well, as it gets them all coming in.
I just read you had a dead chicken still ... I think I would pull some feathers and rub them with liver/blood and spread them out in front of your set to use as chum.
Some times it's a long game when going after a specific critter, because you're educating them the whole time.... I once educated ( and fattened up) a grey fox over the course of a few weeks ... each day my trap was robbed of bait, and I'd adjust... same thing the next day...I never did catch the silly thing.
Are you setting the traps in the coop, in a shed, outside?