Is swimming through spacetime only possible if the spacetime is curved in some way that breaks the symmetry under Lorentz boosts?
It's impossible in Minkowski spacetime, and anything else breaks the global Lorentz symmetry. It doesn't even have to be curved. For example, in the cylindrical spacetime obtained from Minkowski spacetime by identifying $(t,x,y,z)$ and $(t,x{+}1,y,z)$, the curvature is zero everywhere, but you can change your $x$ coordinate by tossing a ball in the $+x$ direction and catching it when it returns from the $-x$ direction. The argument from Noether's theorem doesn't rule this out because this spacetime isn't invariant under boosts (except in the $yz$ plane).
My gut reaction on first reading it was "this violates conservation of momentum, doesn't it?". I now realize, however, that this doesn't allow something to change its momentum; it only allows something to move (change position) without ever having a nonzero momentum.
You can change your momentum by "swimming". Even in Newtonian gravity, if you're at rest in a nonuniform gravitational field, and remain at rest because it happens to integrate to zero over your mass, you can generally change the value of the integral by redistributing your mass, and thereby accelerate. This doesn't violate conservation of momentum because there's a backreaction on the sources of the field. In general relativity, instead of saying that the field is nonuniform, you say that spacetime is curved, but it's a still a gravitational field, and you can "swim" for essentially the same reason as in Newtonian gravity. It's difficult to say what conservation of momentum should mean in GR, but you presumably can define a pseudomomentum that is conserved if you don't neglect the backreaction.
This paper of course neglects the backreaction. In fact, nothing about the paper makes sense to me from a physical perspective. It starts with a discussion of swimming in fluids at low Reynolds number, where friction is so high that inertial motion is effectively impossible, and uses that to motivate a discussion of motion in vacuum, where not only is there zero resistance to motion, but the distinction between motion and rest doesn't even make sense. It's inevitable that you can accelerate and not just change position by gravitational swimming because it's impossible to even distinguish a state of zero gravitational acceleration in a generally covariant way.
The paper by Harte that's mentioned in the top-voted answer also ignores the backreaction, and is effectively invalidated by that. Its conclusion says:
Even in the presence of linear and angular momentum conservation laws, it was shown that bodies can control the magnitudes of their center-of-mass acceleration and spin using purely internal processes.
The bodies he studied are (as he notes) surrounded by, and permeated by, a fluid with exactly the properties of the ideal Hubble fluid. The effects he found are due to gravitational interaction with that fluid. When the body changes its linear or angular momentum, it's imparting an equal and opposite momentum to the fluid. It could propel itself far more effectively in this situation by exploiting a stronger interaction than gravity, using a Bussard ramjet for instance. Even literally swimming in the Hubble fluid would move you by an amount that would certainly be many orders of magnitude larger than the effect that Harte found (though still far too small to be useful).
The basic issue with both papers is that they ignore the Einstein field equations, and just do differential geometry on a fixed spacetime background. The Newtonian equivalent of that is ignoring $F=GMm/r^2$ and just using $F=ma$ and a fixed background force field. Of course you'll find reactionless propulsion in that situation: you assumed it.
This answer is related.
