Would using neurons to transmit and translate information between the processor and the transmitter/receiver get rid of the risk of a virus getting through?
Yes. What you're describing is a technique already implemented without using biological neurons attached networks called "sandboxing". You can create isolated contexts that are only allowed access to the resources set-aside for them. Cellphones use them for apps. Even web browsers use these with CORS (cross-origin resource) limitations. Any application can run in the system, but it is required to live within those limitations.
Many previous answer cited the Halting Problem as proof that it is impossible to absolutely determine the behavior of a system allowing any input. It is true that for a program that allows ANY input, and when you want a absolute solution. But this is, in my opinion, a mistaken application of the Halting Problem. You could cite Zeno's Paradox as proof that you can not cross the room, but would be a similar, in my opinion, mis-application.
If you are willing to disallow certain inputs (perfectly acceptable in a security context), and you are willing to settle for "good enough", you can get pretty good answers.
You can also ring out complex attacks by translating across machine languages : in-and-out of analog signals, in-and-out of some human-readable representation using a limited character set (XML without CDATA).
There is also a security technique called "air gap" security, which is a low tech version of exactly what you propose. In an "air gap" there is a human operator servicing requests sent in on the network. If everything looks right to the human operator about the request, he or she will go to a completely disconnected other system (paper filing cabinet, computer in a different room, or different computer at same desk with all its own peripherals and unconnected to the outside world) to get the answer. Answer in hand, the operator will then type in the answer to question without the outside-connected computer ever coming into contact with the secure systems (not even a USB drive)
As a historical side-note: air gap security is what Mrs. Hillary Clinton defeated when she ordered staffers to hand-copy top-secret data from an air-gap protected secure system to her personal one.
Can biological computers AKA brains not be hacked like computers?
Unfortunately, brains can be hacked in a different, but similar way to computers. "Brain Games" is a good series devoted to showing examples, but there are several other excellent shows.
Social Engineering (hacking the human beings) is, I've been told, part of most successful system breaches.
However, human brains are excellent at pattern recognition and classification. This is why neural networks (which are software designed to mimic the human brain) is very good in this role. Pattern recognition is used in anti-virus software to identify "suspicious" software, and is used on higher-end network routers to identify and handle "suspicious" activity.
It should be noted that a lot of what is being sold as "AI" on the marketplace these days is basic line-fitting services like Microsoft Excel had in the 1980s. Linear regression is different than classification, and classification is different than general intelligence.
General intelligence is a set of software designed to mimic the human brain that also has a few days, weeks, months, years of actual experience being "taught" by a "teacher" (which may be a fast-moving program with questions and answers). A general intelligence, like a human monitor in the same role, is looking at a much bigger picture for anything unusual. General intelligence may include procedures so that it can formulate explanations (hypothesis) for unusual behavior, and test those ideas by making changes, or dig in and do it's own research; consulting a broader database of possible explanations. A general intelligence may even manage it's own "focus" and "attention", performing time management to pay attention to the parts of the problem that appear most likely to yield results. It can also take unilateral action in seemingly dire circumstances (closing a connection or shutting down and isolating a system), or ask for help in urgent situations where it feels it's not going to be able to figure out what's happening in an acceptable amount of time.