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Is there any rules that govern when to use 不 and 非 and 無 and 未 in regards to
the meaning of "not" or "un-"

for instance 非表示, 不満
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word-choice kanji
不 and 非 and 無 and 未 usage difference/rule
It's not that there are rules, it's that the different kanji have different
meanings which can help determine which one is the right choice. You can use
the following definitions as guidelines, but please be aware that there will
be exceptions and differences depending on specific words.

`無【む】` Means "absence" (as in: it was never there).

`未【み】` Means "incomplete" in the sense of not having achieved something (as
in: we still have further to go before we get to the end point).

`不【ふ】` Means "incomplete" in the sense of not yet having reached capacity (as
in: the container is not yet full).

`非【ひ】` Means "error", but when used as a prefix it's more helpful to think of
it as meaning "opposite" (as in: it reverses the meaning of the word).

Hope that helps.
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I have a partial answer for 不 and 無.

For a na-adjective, 不～ can be interpreted as ～ではない

For a suru-verb, 不～ is ～しない

For either a noun or a suru-verb, 無～ can be interpreted as ～がない

(Source: Bojinsha's Intermediate Kanji Book, Vol 1, chapter 2)
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My answer is almost the same as Dave's in principle. I hope it brings
something new to the table by introducing the notion of "transcribing" Kanji
words into Japanese:

When I studied 漢文【かんぶん】 in my high school days, I learned that transcribing
kanji words into full Japanese phrases can help understand them better. Kanbun
is "a Japanese method of reading annotated Classical Chinese in translation"
([Wikipedia](http://en.wikipedia.org/wiki/Kanbun)).

If we combine 表示 with each of the four characters 不・非・無・未, and skip the
details, they get transcribed as:

**不** 表示 → 表示せ不 (ず)

  * isn't, doesn't - mostly negates verbs and adjectives
  * may add unfavorable connotation
  * 「ページ不表示について」 When pages don't get displayed. - [source](http://www.mitsubishielectric.co.jp/mypage/about_ssl_b.html)

**非** 表示 → 表示に非ず (あらず)

  * doesn't equal ..., doesn't belong to ... - mostly negates nouns
  * 「広告を非表示にするプラグイン。」 Add-on that turns off ads.

**無** 表示 → 表示無し (なし)

  * doesn't exist - negates existence = absence
  * may add unfavorable connotation
  * 「カロリー無表示のおにぎりを売っている。」 They sell onigiris with no calorie display. (emphasis is on the absence of display)

**未** 表示 → 未だ (いまだ) 表示せず

  * doesn't/isn't ... yet - negates completeness
  * 「未表示のリンクの色は青にしてください。」 Please set the color of unvisited links to blue.

In theory, if you reverse this process - by finding the right phrase using
either of the four Kanjis that matches your intention - you should be able to
get the correct Kanji compound word.

Note: Definitions are taken from [wikibooks on
Kanbun](http://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E5%8F%A4%E6%96%87/%E6%BC%A2%E6%96%87%E3%81%AE%E5%8F%A5%E6%B3%95#.E5.8D.98.E7.B4.94.E3.81.AA.E5.90.A6.E5.AE.9A)
and [a study material by a Kanbun
enthusiast](http://www.seiwatei.net/kanbun/hitei.htm).

Note 2: In case of 表示, "不表示" and "非表示" don't have much difference in meaning,
although 非表示 is the prevalent one. My guess is that 非表示 is more neutral, and
therefore more versatile and widely used.
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_Despite the question seems to have reached its own solution, I'd like to make
a new answer, for I find the existing explanations would not lead to a correct
understanding on this topic._

* * *

Each of these words has their own meaning, which usually cannot be directly
translated to English affixes like "non-", "in-", "un-", or "-less", so I'm
going to explain these terms descriptively.

What they have in common is negative meaning and ability to attach to 漢語
([Sino-Japanese](https://en.wikipedia.org/wiki/Sino-Japanese_vocabulary))
words.  
(I didn't consider much about two-character compounds like 不安, 非道, 無常, 未婚 etc.
because these words are better memorized en bloc.)

### In principle

  * `不～` (read ふ) 

    1. makes **state** from **action** , the outcome is usually a **noun**  
"not to _action_ ", or "no _action_ -ing"

> 不使用【ふしよう】 "disuse", 不変化【ふへんか】 "invariable(ness)", 不合格【ふごうかく】
> "rejectable(ness)"

In this usage it's _tenseless_ negation, that is, `保存料【ほぞんりょう】不使用` "No
preservative used" doesn't only mean "unused" but "never be used". Some
speakers may also treat this as a formal way of saying `～しない（こと）`.

    2. makes **state** from **state** , the outcome is usually an **adjective noun** (形容動詞)  
"not being _state_ ", "being low on _state_ "

> 不誠実【ふせいじつ】 "insincere", 不人気【ふにんき】 "unpopular", 不健康【ふけんこう】 "unhealthy"

Almost limited to subjectively undesirable state.

Can also attach to native words.

> 不まじめ "lazy", 不[行]{ゆ}き届{とど}き "mismanagement", 不ぞろい "uneven"

  * `非～` (read ひ)  
makes **identity** or its **holder** from **state** , the outcome is usually a
**noun**  
"not belonging to _state_ " or "being other than _state_ " or "non- _state_ "

> 非公開【ひこうかい】 "non-disclosed", 非会員【ひかいいん】 "non-member", 非現実【ひげんじつ】 "irreality"

Maybe better explained as [logical
NOT](https://en.wikipedia.org/wiki/Logical_connective). If only two options
are in question, it's equal to "un-".

Can attach to any kind of words.

> 非ユークリッド "non-Euclid(ean)", 非[割]【わ】り込【こ】み "non-interrupting"

  * `無～` (read む)  
makes **state** from **thing** or **action** , the outcome is usually a
**noun**  
"without _something_ " or " _something_ -less"

> 無重力【むじゅうりょく】 "weightless(ness)", 無公害【むこうがい】 "pollution-free(ness)",
> 無所属【むしょぞく】 "independent (of a candidate)"

Can attach to any kind of words by theory, but other means are preferred
outside 漢語: `底{そこ}無{な}し` "bottomless", `バリアフリー` "barrier-free"

  * `未～` (read み)  
makes **state** from **action** , the outcome is usually a **noun**  
"not yet _action_ -en"

> 未確認【みかくにん】 "unconfirmed(ness)", 未成熟【みせいじゅく】 "unripe(ness)", 未解決【みかいけつ】
> "unsolved(ness)"

This one is preferred when the action is considered
[telic](https://en.wikipedia.org/wiki/Telicity) (i.e. has a goal), even if you
do _not_ really have intention to achieve it.

**Extra**

  * `不/無～` (read ぶ)  
makes **state** from **anything** , the outcome is usually an **adjective
noun**  
"ill- _something_ " or "mis- _something_ "

> 不器用【ぶきよう】 "clumsy", 不細工【ぶさいく】 "ugly", 無愛想【ぶあいそう】 "unamiable"

Though of high frequency it may be seen in, it's no more productive (i.e. able
to make new words using it), and technically it's less related to those ones
above, for attaching only to those almost-nativized 漢語 words.

### Actual usage

However, existing words with these prefixes often betray your expectation on
how it works, mainly because:

  * Bare 漢語 words are by nature flexible in grammatical status, as like in original Chinese, as well as some English words (round (n), round (v), round (adj), ...). You should fill in implicit participlization. That's why I avoid to use "verb" or "noun" etc. in the list above.  
e.g. `非表示【ひひょうじ】` → "being-other-than display **ed** " = "undisplayed(ness)"

  * In general, established nouns that mean qualities easily turn into adjective(-noun)s in Japanese. Many common words which should be "noun" according to my list, are actually adjective nouns.

  * Modern Japanese (Meiji-WWII) embraced many words in different word classes than today's. Besides, words themselves could have undergone changes regardless to its original meaning.  
e.g. `不経済【ふけいざい】` → "diseconomy" or "being-not econom **ical** " < `經濟` _ModJ_
"economize"  
`非常識【ひじょうしき】` → "non-common-sense" → "lacking in common sense"

Samuel Martin's _A Reference Grammar of Japanese_ has [a good summary and
discussion](http://books.google.com/books?id=SszxbMtHbs8C&pg=PA388#v=onepage&q&f=false)
on those prefixes, but note that he has erroneous analysis on some words
regarding the said matters.

### Quick chart

`会員` as an identity (to be a thing), `現実` as a thing as well as state, `健康` as
a state (adjective), `表示` as a state as well as action, and `使用` as a pure
action.

```

    　　　　不　　　　非　　　　無　　　　未
    会員　　×　　　　○　　　　○　　　　×
    現実　　×　　　　○　　　　△　　　　？
    健康　　○　　　　○　　　　×　　　　？
    表示　　○　　　　○　　　　○　　　　○
    使用　　○　　　　△　　　　○　　　　○
    
```

**Acknowledgement** : Thanks to @snailboat for references and advice.
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