Would a swivel wheel trailer work with a bicycle? I want to use a trailer with my bike but it's extremely dangerous in winter conditions due to black ice and slush. 

I'm looking for a trailer that acts like 2 extra wheels for stability than just something to drag. So imagine a trailer box that is hard bolted to the bike frame so the trailer always keeps the same angle of the bike frame. The 2 wheels on the trailer pivot. 

Have you ever seen this done? 
Do you think it can be done with a bike?
Would it work? Sure, technically. You'd have to fashion one yourself (or have it made for you). If you want the frame to stay fixed and just the wheels to swivel, unlike a chariot, then you'll likely need to weld the frame to your bike, making a strong commitment to winter weather (and permanently increasing its footprint in your garage).

Will it be comfortable or beneficial to your ride? No. In order to turn a bicycle, the vehicle leans. Without leaning, your inertia will play a big part in the direction that you go; i.e. you're going to have to take it slow around corners to avoid falling over. You'll need to slow down to a speed where you'd be unlikely to fall on black ice, provided you have appropriate tires, making your trailer redundant.

In my experience, riding year-round in all weather from freezing rain to a foot of snow to a tornado (oops), well-treaded tires are usually good enough, provided you pay attention to the road conditions. At times, it turns the activity into something more akin to skiing. If you wanted to really make sure you don't get surprised by ice, studded tires are the easiest answer, though if you're cycling in all kinds of weather, it can be a PITA to have to switch between studded tires and treaded tires once the ice clears.

To summarize: yes, you could do it, but it's not worth it.