Will a BigEasy driver handle 14 amps?
The Big Easy driver is based on the Allegro A4983 chip, which can handle only up to 2 A per phase. So no, it will definitely not handle 14 A (even if you use both of the internal H-bridges in parallel), and the chip's built-in overcurrent protection circuitry will prevent you from consuming anywhere near that much current.

But for your situation, a stepper motor controller is way overkill for toggling solenoids. A mechanical relay, an SSR, or even a MOSFET of sufficient current capability will be more than adequate to control the electromagnet.



Are you sure you have a 24 VDC solenoid that consumes 14.6 A? That would mean that the coil resistance is only 1.64 ohms, which seems really low to me.