I was reading the wikipedia page on "predicate," where it mentioned that in Japanese, the negative form of a verb is an adjective. I thought that this was too expansive a statement to be true, but what does the writer mean? Does the writer mean that only verbs in the ない form of the negative are adjectives? Is this still too broad in meaning to be true?
No, they aren't adjectives.
They mostly follow the same basic grammatical rules (with a few exceptions) that い adjectives do.
However there are several grammatical constructs that only with either ～ない verbs or い adjectives.
More over in classical Japanese ～ない was things like ～ず ～ぬ ～ん most of the time. Those constructs have zero resemblance to い adjectives. So anyone attempting to argue that ～ない is effectively an adjective would have to argue that negative verbs became adjectives only in modern times.
Most importantly Japanese text books clearly classify them differently. 動詞の否定形 and い形容詞.
In addition ～ません has very little relevance to this question because ～ます is less of a verb form and more correctly the continuative(い ending) form of a verb + an irregular verb.
Modern Japanese is very different from archaic Japanese (and some modern formal written Japanese, which is itself rather archaic) in regard to the topic at hand. Initially there were distinct conjugations of verbs and adjectives known as predicative and attributive. Predicative (also called conclusive) was used for the final verb in a sentence, and was finite. The attributive form was used nominally and for relative clauses (remember that technically "adjectives" in Japanese are relative clauses), and seems to be non-finite. Rather than the modern -い ending of adjectives, adjectives had the ending -し in predicative form and -き in attributive (for modern ない that would give なし and なき, respectively).
For an example of this, and an example that shows this is still in use in written Japanese, take the title 藍より青し. Here, 青し is the predicative form of the adjective, with the original -し suffix.
At one point Japanese underwent a radical morphology change, and actually lost the predicative form entirely, completely replacing it with the attributive (at least in form; the semantic distinction between finite and non-finite remained). At this point a large number of old verb suffixes disappeared. The old negative suffix, for example, mostly disappeared, replaced by -ない (at some point the attributive -き suffix of adjectives was reduced to -い, as well). Presumably the old suffixes required finite verb forms, and were lost when the morphologically finite forms disappeared, replaced by the morphologically non-finite forms that had the same meaning (i.e. -ない).
Now, I imagine when you heard that negative verbs were adjectives this was based on the fact that ない is the head of the verb phrase for negative verbs and it's an adjective. This is indeed a change that occurred at one point, and was not always the case, as I spent the majority of this post explaining. My hypothesis is that the old suffixes used with predicative verbs were themselves predicative (auxiliary) verbs, and the old attributive and modern suffixes are in fact adjectives or nouns (or derived from them).
Arguing about whether certain words "are" something or other is missing the point in this context, I think. We do not classify words based on some innate, a priori nature that we discern within them. We classify them based on behaviour. And there is no a priori set of standards for that classification either: we have to choose our own. It's completely arbitrary. But if we aim to be scientific, we will generally want to choose standards that are as simple as possible while explaining as much as possible (with as few inconsistencies and exceptions as possible).
So, take the polite nonpast negative ending -nai. It ends in -i, like an adjective. It takes endings like -katta and -kereba like an adjective. It can be used adnominally or conclusively, like an adjective ("shiranai hito" vs "ano hito wa shiranai"). Particles like "kedo" and "shi" attach to it like an adjective. (Both of these last two points apply to verbs too, of course.) On the other hand it can't be passivised like a verb.
In short, we can list many ways in which it acts like an adjective, and many ways in which it doesn't act like a verb. On that basis, we might decide that it is simpler to just call it an adjective than a verb-in-disguise. We would advance a theory in which we have a category called "adjective" that includes words like "akai", "hiroi", "kanashii" but also "ikanai" and "shiranai".
On the other hand, objections can be raised. In this thread, Ian has raised some. I find them unconvincing (for the reasons given in dainichi's comments), but we can find other differences. For example, the "-ide" ending does not seem to be available to "natural adjectives": we can say "ikanaide" but not "akaide" or even "akakunaide". So we have to adjust our theory to account for this. Maybe we say "There are two kinds of adjectives, one natural and one derived from verbs via the -nai ending; they behave very similarly but we observe these differences..." Or maybe we say "Verbs in the negative nonpast remain verbs; they exhibit some very adjectivelike behaviour but we can tell the difference because..." We have to decide which theory fits the facts best, ideally in the sense of "best" described above (simplest, most powerful, etc.)
Now, all of the above said, let's look at the -masen ending. It doesn't seem to behave much like an adjective at all, except in very trivial ways So even if we define our category of "adjective" such that it includes words with the -nai ending, we probably won't put words with the -masen ending in there too. We could -- it's arbitrary! -- but doing so wouldn't help us explain very much. It would just make our theory more complicated. So, we don't.
Thus, what I think the wiki editor whose description you read meant was "once the plain nonpast negative ending -nai is attached to a verb, the resulting word behaves so much like an adjective that it makes more sense to classify it as one (in the classification scheme I am implicitly adopting)." Not "any negative verb phrase is an adjective by definition," or anything extreme like that.
Dainichi's comment on your question is another example of the principle that we are free to set up our categories in any way we like. Adjectives and verbs are obviously different in many ways, but there is also a sense in which they are similar. Given the structure "Hana wa X", a verb like "saku" could fit in there, or an adjective like "akai". They fit into the same "slot" like that. This isn't true of English; we can say "the flower blooms" but not "the flower red"; we have to say "the flower is red" (and, incidentally, for this reason most linguists working on English will recognize a category that contains both "blooms" and "is red").
So in that high-level sense, Japanese adjectives and verbs form a unified superclass of words, one that can be distinguished easily from, say, nouns, or particles. That isn't any less true just because we can subdivide that superclass into "verbs" and "adjectives" based on morphology. There might be cases where using this superclass makes things easier for us (e.g. looking at high-level syntactic patterns), and there might be cases where it's easier to keep verbs and adjectives distinct (e.g. looking at word endings). Neither analysis is wrong; they're just different tools for different tasks.
As others have argued, it's pretty much a question of definition. But it seems obvious that in form, there's overlap between i-adjectives and plain verb-negatives.
I'm going to try to be constructive about it by starting a list of forms that exist in either or both of the cases. Please add or comment (or fix my formatting) as you see fit.
The correspondence seems pretty good to me. And the slight problem with なくない, I consider inherited from ない, since 無くない itself is considered wrong by some.