Which version is proper: "neither is good" or "neither are good"?

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In fact, there is a fault in reasoning from the first line of your argument: 'Since you are using "neither", you must be talking about two things, so you should use "are".'

"Neither" in fact makes reference to only one of the two options. Consider the example "neither of us is hungry". It could be explicitly expanded as "neither one of us [...]", therefore demanding the singular. The view that it is plural is common (in this case, "us" is the impetus), but it is not Standard English.

At the end of the day, what matter most in this scenario are the needs of the person who asked this question. In formal contexts, perhaps it should be consistently considered singular, but colloquial usage allows for more leeway.