I've got some RG59 and RG6 snakes that need the BNC ends replaced. RG59 compression ends are working perfectly - the RG6 black outer shield is so tough I can't get the ends on no matter how hard we push.
It's quad shield, and I've stripped everything away to create my 1/4: 1/4 strip.
I've tried 3 different brands of compression end, including what the cable manufacturer recommends.
If the connector doesn't fit, you are using the wrong connector for that particular wire. That quad-shield stuff is often used for RF Cable distribution, and not that often for video. So most BNC connectors aren't made for it, and F connectors are more common on it. That is not to say you can't find a BNC that will work, but it is something you will have to do some searching for.
I highly recommend you change coax instead of connectors, though. Belden 1694A is an awesome coax, and is very easy to find good quality connectors for. It is also better at handling HD-SDI video than quad-shield. Most quad-shield, as I said, is for RF applications, is comparatively cheap, and often uses copper covered steel instead of solid copper. It is made that way to be cheaper, and to handle the stress of cable installations - which might include things like flying the cable from a telephone pole to a house. It isn't as good for video. And honestly, it isn't even as good for RF cable. I replaced my Cable lines in my house with 1694-A, and gained 6 dB on my cable signal on the far side if my house. That steel simply doesn't work as well as copper.