So per John H, you should be able to run the lights with DC. Based on this OGR thread, you need to put DC+ on the red wire. That is, for the technically curious, it appears the crossing lights PCB only has a single diode to convert AC to DC (vs. a 4-diode bridge-rectifier); hence you need to apply DC with the proper polarity.
As for as volume control, as GRJ says, you can splice a variable resistor (aka potentiometer) into one of the 2 wires going to the speaker. But let's be absolutely clear; the 2 wires going to the speaker are INSIDE the black chamber. The 2 wires going to the black chamber itself are the external AC (or DC) power wires. You cannot just splice in the potentiometer to an EXTERNAL power wire going to the chamber. I believe discussion of muting the speaker with a simple on-off switch is referring to simply cutting power to the chamber (EXTERNAL wire).
So. If your head isn't spinning yet, for the most flexible volume adjustment you need to open the chamber per the linked thread. It will look like this picture from the other thread:
Then splice in the potentiometer (GRJ gave a suggestion) to one of the 2 wires going to the speaker.
To make matters more confusing, if the $6.60 (plus shipping) of the 100 Ohm potentiometer is too much buck-for-the-bang, you could use a 3-position toggle switch (aka SPDT ON-OFF-ON) to swap in a fixed resistor (more like 10 cents) and simply have low-OFF-high volume selection.
