IIUC a negative vote would subtract a vote from a candidate instead of adding. In theory, if enough people vote negatively for the two big parties, a third one could arise and perhapas surpass them after gaining enough traction.

And how is that different from just voting for a third party?

It does solve the issue of being strictly against a policy of one candidate and having no consolidated view for others.

Imagine Alice, Bob, and Eve are running for president. My main concern is keeping is keeping Eve out. As for Alice and Bob, they have their pros and cons, but at the end of the day I don't care too much either way and both are much better than Eve. In this scenario a negative vote for Eve would be more in line with my desires than flipping a coin and choosing between voting for Alice or Bob.

Ranked choice already solves this and as a bonus you get to pick whether you like Alice or Bob better.