In particular this one Cobra Products 25Ft Power Drain Snake.
Will a 12-volt cordless be strong enough?
Depends on the size/strength of the clog.
However, as an owner of a Hitachi 12v cordless, I find it quite strong as long as the charge is good, so probably.
According to customer service for this manufacturer, yes, 12v is fine.
YMMV for other comparable products