Can a magnets stop a speaker from moving?

That depends firstly on what kind of speaker you have. If it is piezoelectric, then it would most likely have no effect.

If it is electrodynamic, then it would be affected, but unlikely to the point of muting it entirely. Such speakers already have very strong magnets, the field of which is modulated by a coil. Add another magnet, and the modulation is unaffected as such, although the magnetic filed would be distorted which could affect the sound in terms of pitch and resonance, but is unlikely to quiet it.

What would work would be for the supplemental speaker to be positioned such that its filed overlaps the existing one, and be fitted with the coil driven by a modulating current that is exactly opposite to the one provided to the speaker.

For a normal moving coil speaker, no. There is no dc current in the coil, it's just the ac audio signal. So there is nothing for a magnet grab onto apart from a field due to the signal, which keeps alternating direction.

No