So these "particles" are actually not particles, but "quasi-particles".

In other words, throw away scientific rigor as soon as you can get a clickbaity headline.

"When the quasiparticles travel along one dimension inside the ZrSiS crystals, they do so at the speed of light and are therefore massless. But as soon as they try to travel in a different direction, they hit resistance, slow down and gain mass."

It's like traveling from vs. to the US and Europe and yes, those planes also gain/lose infinitesimal amounts of mass.

Could they create a non-Newtonian drive with this?

Let the particle race around a circular track, and you'll have a net force in one direction.

https://en.wikipedia.org/wiki/Reactionless_drive

It's not a particle, it is a quasi-particle. Those are field manifestations that happen on solid state, like solitons and holes, so it is cool, but no fundamental change do the standard model or anything like that.

I think “loses” lends itself to clickbait.

It’s either 0 or massful.

“Loses”, to my eye, implied a continuous ability to remove mass, and led me to wonder if one could achieve negative masses with the experiment’s technique.

But I’ll save you the click: in one direction it’s massless (traveling at the speed of light), and in the other direction it has mass.

*quasi-particle