The short answer is: Yes, it is true, because the contrapositive just expresses the fact that $1^2=1$.
But in controversial discussions of these issues, it is often (but not always) a good idea to try out non-mathematical examples:
"If a nuclear bomb drops on the school building, you die."
"Hey, but you die, too."
"That doesn't help you much, though, so it is still true that you die."
"Oh no, if the supermarket is not open, I cannot buy chocolate chips cookies."
"You cannot buy milk and bread, either!"
"Yes, but I prefer to concentrate on the major consequences."
"If you sign this contract, you get a free pen."
"Hey, you didn't tell me that you get all my money."
"You didn't ask."
Non-mathematical examples also explain the psychology behind your teacher's and classmates' thinking. In real-life, the choice of consequences is usually a loaded message and can amount to a lie by omission. So, there is this lingering suspicion that the original statement suppresses information on 0 on purpose.
I suggest that you learn about some nonintuitive probability results and make bets with your teacher.
First we need to assert the general framework. We have a language with relation symbols and function symbols and constants, etc. With this language we can write sentences and formulas.
We say that $T$ is a theory if it is a collection of sentences in a certain language, often we require that $T$ is consistent.
If $T$ is a first-order theory, whatever that means, then we can apply Goedel's completeness theorem and we know that $T$ is consistent if and only if it has a model, that is an interpretation of the language in such way that all the sentences in $T$ are true in a specific interpretation.
The same theorem also tells us that if we have some sentence $\varphi$ in the same language, then $T\cup\{\varphi\}$ is consistent if and only if it has a model. We go further to notice that if we can prove $\varphi$ from $T$ then $\varphi$ is true in every model of $T$.
On the other hand we know that if $T$ is consistent it cannot prove a contradiction. In particular if it proves $\varphi$ it cannot prove $\lnot\varphi$, and if both $T\cup\{\varphi\}$ and $T\cup\{\lnot\varphi\}$ are consistent then neither $\varphi$ nor $\lnot\varphi$ can be proved from $T$.
When we say that CH is unprovable from ZFC we mean that there exists a model of ZFC+CH and there exists a model of ZFC+$\lnot$CH [1]. Similarly AC with ZF, there are models of ZF+AC and models of ZF+$\lnot$AC.
Now we can consider a specific model of $T$. In such model there are things which are true, for example in a given model of ZF the axiom of choice is either true or false, and similarly the continuum hypothesis. Both these assertions are true (or false) in a given model, but cannot be proved from ZF itself.
Some theories, like Peano Axioms treated as the theory of the natural numbers, have a canonical model. It is possible that the Goldbach conjecture is true in the canonical model, and therefore we can regard it as true in some aspects, but the sentence itself is false in a different, non-canonical model. This would cause the Goldbach conjecture to become unprovable from PA, while still being true in the canonical model.
Footnotes:
- Of course this is all relative to the consistency of ZFC, namely we have to assume that ZFC is consistent to begin with, but if it is then both ZFC+CH and ZFC+$\lnot$CH are consistent as well.
Best Answer
These statements taken together are called inconsistent. That means that they cannot all be simultaneously true. But the first and second are neither true nor false without broader context. Using the language of first order logic, they might be said to be formulas with "free variables." Here's an example of a formula with free variables $$ 4x+3y=9$$ This is neither true nor false because I haven't told you what $x$ or $y$ are. If I use quantifiers to get rid of all the free variables, then I have a sentence which may be true or false: $$\forall x\forall y (4x+3y=9)$$ $$\forall x\exists y (4x+3y=9)$$ $$\exists x\exists y (4x+3y=9)$$ The first statement is false, while the second two are true.