I keep most of my sealants and liquid waxes in a cooler in the winter. It helps keep them from freezing. In fact, if you keep them in any kind of cupboard in your garage then you can usually prevent them from freezing, but I have been there before where that just isn't an option. Bring them indoors if you feel it is going to be too cold. Nothing sucks worse than a frozen bottle of $50 sealant. lol Trust me....
So I typically use soapy water for clay lube unless I buy clay that comes with it's own spray. I just haven't found the value for paying money for lube when soapy water works just as well. Yeah it is a little messier, but the next step is polishing the paint anyway, so who cares about a couple soap streaks when it is all said and done? A quick rinse of the car after you are done claying gets rid of most of that anyway. Actual clay lube is nice, but simply isn't worth the money. It's not like the clay picks up stuff better if you use it or anything like that.
You can dip an empty (or re-purposed bottle from another product) into your wash bucket before you wash your car and that is just the right amount of soap for clay lube. Do it before you wash the car though.
No sense in using dirty water. You could also just squirt a little bit into a bottle. Leave a little air at the top so you can shake it up, but it tends to do that pretty well all by itself. Use too much soap and it will just coat your clay and eventually disintegrate it, so go easy on it.