Suppose that any statement like these is proven to be undecidable or independent of ZFC. Wouldn't that be the same as proving it to be true?Because if it's undecidable or independent we can never find a finite counterexample, so it's safe to assume that it's true.
In general, a statement of the form "there is no [thing] such that [fact]" is true if it is (a) un-disprovable from (say) ZFC,
and (b) is reasonably "simple": the "things" have to be things we can search through in an "easy" way (like natural numbers, and unlike arbitrary real numbers), and the "fact" has to be something which is "easy" to check (like "is prime," and unlike "has the property that there are infinitely many primes with that difference").
These, ultimately, turn out the be the $\Pi^0_1$ sentences - this is a notion of complexity coming from both proof theory and computability theory. Or rather, they are the sentences equivalent to some $\Pi^0_1$ sentence.
At this point it's good to see some examples:
The Goldbach conjecture is manifestly $\Pi^0_1$.
The Twin Prime Conjecture is not (so far as we know) equivalent to a $\Pi^0_1$ sentence: a "counterexample" to TPC is an $n$ such that there are no twin primes above $n$, but "there are no twin primes above $n$" is not an "easy to check" property.
Finally, there are really interesting situations where something that (a) doesn't look $\Pi^0_1$ and (b) isn't known to be provable is still equivalent to a $\Pi^0_1$ sentence. The Riemann hypothesis (RH) is a great example of this: it is not obviously $\Pi^0_1$ - a counterexample to RH is a complex number, and there's no easy way to search through all the complex numbers. However, it turns out that we can prove in ZFC that RH is equivalent to a certain $\Pi^0_1$ sentence - see this question - and so (in a surprising way!) has the property that, if it is un-disprovable in ZFC, it is true.
Incidentally, there's nothing too special about ZFC here. The only property of ZFC being used to show that $\Pi^0_1$ sentences not disprovable in ZFC are true, is the fact that it is $\Sigma^0_1$-complete: if $\varphi$ is a true $\Sigma^0_1$ sentence, then ZFC proves $\varphi$. Much weaker theories have this property - even theories much weaker than PA!