The bunnies I skin and gut are either snowshoe hare or the domestic rabbits I breed for meat.
The snowshoe hare I either shoot or snare. If I shoot them I squeeze out the urine right away by holding the bunny against my legs head up (...me and the bunny...
. I then use my round fist and press it in a downwards motion down their belly. They start peeing right away...
If I snare them in the winter they are frozen stiff by the time I find them. No squeezing possible...and sometimes you can taste/smell it. I had one that smelled horribly like urine while cooking.
I skin first, then do the gutting after. I hang them upside down, quick incision around the ankle, then cut skin all the way legs down (inside of legs) to genitals on both sides. Pull skin down the legs (sometimes a little cutting needed), separate skin at the back with hand only, cut skin behind anus (incl. tail off), then pull skin like a sock all the way down past the head and down the arms. I cut the head off and the hands at the wrist joint. Now you have the naked bunny hanging in front of you. I then cut the belly (only the flesh, do NOT puncture guts) open, guts fall out, rip open diaphragm and remove heart & lungs. When removing liver, make sure you do not tear gall bladder (the little yellow sack attached). If you want to use liver you can pinch the gall bladder and rip it off the liver. Just make sure you do not spill any of the liquid. Some of the intestines will still hang attached through pelvis and attached at anus. I spread the bunny legs as much as possible and take a sharp knife to cut the pelvis open. There is a spot on either side a little off-center where you can go through easily. Once that is open you can remove the remaining intestine. Bunny is now skinned and gutted. The two kidneys are still attached along the back in the abdomen. You can leave in or take out - depends on you.
It only takes a few minutes to process a rabbit this way, so I never bothered squeezing urine from my domestic bunnies. Never had issues.
The skin on snowshoe hare is so fragile, you need no knife to tear it open for skinning. Couldn't believe it until I tried the first time.
Another curious thing about killing rabbits and the meat quality: The first domestic rabbits I killed, I shot in the head. The meat was terribly tough. A friend suggested that was because of the cramping muscles after the head shot, as he had experienced the same with a moose. The next batch of rabbits I killed by stunning with a blow to the neck and then bleeding. The meat was very nice and tender. Anything else in the process of skinning, gutting, freezing, cooking was the same. So I assume he was right. No more head shots for me...I mean for my rabbits, of course.