I am kind of disturbed by the idea of cryonics.

Why?

Wouldn't it be theoretically possible to prove they don't work, assuming that they really don't.

If it cannot work, then we would expect to find evidence that it cannot work, yes. But it sounds like you're starting from a specific conclusion and working backwards. Why do you want to "prove [it doesn't] work"?

If the connections between neurons are lost in the process, then you have died.

Alcor's FAQ has some information on the evidence indicating that cryonics preserves the relevant information. That depends on the preservation process starting quickly enough, though.