Burlap would definitely keep the snow out.You could also use something like landscape fabric or black plastic.Two open ends will vastly increase your odds of a cat entering,as well as aligning the cubby in the direction the cat already wants to go.

If you are on a 24 hour check,snow shouldn't be a problem.Just sweep it off the traps after each storm.Generally cats hole up during a storm and move the next evening.When you clear them off,kick back the snow away from the cubby entrance,making a big sight attraction with leaves and top soil that stands out at a distance.It can be quite a draw.

I just watched and episode of Trapping Inc (2019 season episode 12) and Rich Mellon makes a cage from woven wire but he turns his up on its end and stows it under a cedar tree so it cant be entered from above. He is up in Canada so he does cut and opening for a 330 but that is for the Wolverines.

For Lynx cats he prefers to block in and snare trails going both into and around his home made cubbies. I had never seen it done like that nor would I have ever thought to do it that way. Rich, or course, was loading the sleigh high with cats!

That's a common set for lynx which I used for years. A double end snare cubby with snares hung in all tight spots around it.
Often double or triple catches.

i have used white round buckets, white square buckets, Green square buckets, chicken wire wrapped with pine bows, old stump wood stacked to make a cubby and have caught cats in all.
My thinking has changed since i just started using footholds with some flagging and scent. This appears to get larger cats. I think the large males shy away from cubbies.
So that is my preferred way now. No cubby and just footholds.

Once you learn to get rid of the roof on cat cubbies,your catch will go up substantially-and set like tomcats set up for multiple cats per check.

I've been meaning to ask you, do you set "on sign" or do you set in likely looking habitat? I set on sign and even then it can be a real waiting game for the cats to make their rounds.

I dont always set on fresh sign,but I know where the cats travel and hunt,and have seen plenty of sign in these places year after year.I will make sets in these spots even if there is no fresh sign.
Setting on fresh sign where cats have stopped to hunt for a few days is very productive usually with quick results.
Setting on travel routes is a waiting game,but they eventually produce when a cat or family of cats comes thru.Setting on fresh sign along a travelroute is of no benefit,as the cats have already passed.You want your sets working when they come by on their way to another feed pocket.
With a snowmachine in winter,It doesnt take long to scout out and get to know the feed pockets and travel routes of the cats when they are plentiful on the landscape.Once identified you can count on these spots to produce year after year.After a few years the real productive ones will stand out and the hit and miss spots can be weeded out.

I set in good hare habitat with considerable activity where the lynx will be hunting, using a visual attraction to draw the cat close to the set where the lure or bait draws them in to investigate.

Set up on locations and tracks cubbies can be used as an attractant you can blind set the cubby 5' to 10' out,with natural walk throughs etc.that way the bobcat does'nt have to commit,plus you can pick up other critter's Boco calls them Jack pots,i call them Mine Fields.Many times i might have 6 snare's or footholds surrounding the cubby set from 20' to 50' out,it works WELL!!!!!!!!!!!!Good luck.

This is great advice. I was doing this also. I've had a dozen plus snares within 20 feet of a cat set. I bought a bunch of bright colored clothes pins and put them on the trees and brush where the snares were, and wrote down a count of traps at each spot. I can check a cat set in seconds without having to remember where all the snares are, just a quick count of the pins, and a visual of the snares.

I'm going to try some cable restraints this year. My season starts 12/26