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Thermodynamics doesn't say that "everything is bound to break down, and that we won't be able to predict all of the ways it can happen".

I guess you're referring to the second law of thermodynamics. It says that the entropy of the universe tends to increase. This is predictable and the lack of "negentropy" won't be an issue for a long time. The heat death of the universe is supposed to be on the order 10^100 years from now, so is not very relevant when talking about curing aging.

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I suspect they were referring to the ways our body is part of numerous pseudo-closed local systems, and in each of those systems entropy will tend to increase. It's true that we are also open systems in meaningful ways and can be farmers of enthalpy inside our bodies. To some extent that's exactly what life is--enthalpy farming equipment. But farming is tricky and requires constant energy inputs. And it does seem like fighting the entropy of those pseudo-closed (ie can be opened with effort) systems is a difficult if not losing battle.

Thermodynamics doesn't say that "everything is bound to break down,

It's pretty close to that.

and that we won't be able to predict all of the ways it can happen".

So either you don't understand how to generalize the Halting Problem, or you didn't get the reference.

I guess you're referring to the second law of thermodynamics. It says that the entropy of the universe tends to increase.

Exactly. There are far more states of a given machine that are broken than there are states of a given machine that "work."

EDIT: Also, to expand on what erikpukinskis says in the cousin comment, you can enthalpy farm to fix what's wrong in any given body. The problem is knowing what to fix. The body is so vastly complex, it's going to be very hard to guarantee we always know what to fix ahead of time for everybody. (Halting Problem) Hence, some small fraction of any population is always going to be dying from unforeseen failure modes, unless some philosophically interesting tinkering takes place. (In any case, it's inevitable that all Homo sapiens will be gone, eventually.)