I don't think it has to do with them being verbs, and for the most part, I also don't think it has to do with the etymology. I think it mainly has to do with the stress patterns of these words.
The tendency I see is that the letter L is generally doubled after a stressed short vowel, and not doubled after an unstressed vowel. Historically, there has been a lot of variation in the spelling of words like this. Based on etymologies that I have looked up, I would say there is little to no etymological significance to the use of a single L vs. a double L at the ends of words. It's just an oddity of our modern spelling system. (Another relevant question about double L: Why is there a double "ll" in "bell"?)
Here are some actual examples: mill (as you've found out, it comes from Old English mylen), small (comes from Old English smæl), skill (comes from Middle English skele, scele, skile, skyle, from Old Norse skil), dill (from Old English dili, dile, dyle). On the other hand, metal comes from Latin metallum with a double L (although the French spelling metal may have also played a role, as French seems to have been an intermediary in the transmission of this word). I have some more examples of words like metal in my answer here: Adding an L when appending an -ium suffix to a word? (Metallium vs. Metalium)
Monosyllabic content words always end in a stressed syllable, and polysyllabic verbs are more likely than nouns to end in a stressed syllable. These two factors are probably why it seemed to you that this was related to part of speech.
There are many nouns that end in "-ill," especially monosyllables such as bill, hill, sill, and frill. Adjectives such as still can also have this spelling.
There are also verbs that end in "-il." Aside from bedevil, there are cavil, imperil, stencil, fulfil (British spelling; Americans use fulfill). However, it does seem to be true that fewer verbs than nouns end in "-il."
What is the etymology of words that end in -il?
It's worth noting that there aren't that a huge amount of words in total that end in -(consonant)-il."
As you've found, many of them are derived from Latin nouns ending in -illus, -illa or -illum. When Latin nouns are taken into English, final -us, -a and -um are often simply removed, which explains how we get English nouns ending -ill or -il.
Latin verbs, however, are usually borrowed in a more complicated fashion. For complicated historical reasons, Latinate verbs in English are generally derived from the past participle form of the Latin verb. Latin past participles generally have a dental suffix (usually t) that is retained in the derived English verb. So for example, the Latin verb inficio corresponds to the English verb infect. In regular verbs of the first declension (the most common type) this dental suffix is preceded by the vowel a. So Latin verbs containing -ill- have generally been borrowed as English verbs ending in -illate. (For example, scintillate comes from Latin scintillare). That might be a contributing factor to the relative rarity of verbs ending in "-il."
There are various other sources for words ending with "-il." In most cases, the modern spelling with "il" is fairly straightforward based on the etymology.
However, there are a handful of words that seem to have acquired the spelling "-il" for unclear reasons. Here's what I mean: devil (Old English deofol), evil (from Old English yfel), weevil (Old English wifel), stencil (earlier spelled stancel; interestingly, the Oxford English dictionary says that the noun is derived from the verb). I don't know how to explain these.
Best Answer
I think part of the premise of this question is incorrect: build did not gain a "u" between Middle English and Modern English. Modern English spelling is standardized, so we can talk about "the" spelling of most specific words. But in Middle English, there are usually multiple attested spellings for the same word; this is partly due to overall lack of standardization (some writers might even spell the same word several different ways on a single page), and partly due to the existence of conflicting standards (Middle English is a cover term that encompasses many different regional varieties over a pretty wide range of time, traditionally from the Norman Conquest of 1066 up until the Great Vowel Shift around the 1500s).
Dictionaries, such as Wiktionary, need to choose one written form to use as the headword (in this case, "bilden"), but this doesn't mean that no other forms existed during this time period. According to the Oxford English Dictionary (OED), forms of this word spelled with "ui" or "uy" actually date back to Middle English, but their origin is unclear:
My guess is that it has something to do with phonetic rounding, as the letter "u" is generally associated with rounded vowels. This could be due to the preceding rounded consonant /b/, the historical rounding of the root vowel (Old English y was the rounded counterpart of i) or some combination of both. I'm not at all sure of this, because as the OED says, the regular development of Old English y is complete loss of rounding and merger with i, as in words like kiss v. (from Old English cyssan v.) or lice n. (from Old English lȳs n.).
However, modern standard English vocabulary does also show some influence, in both pronunciation and spelling, from dialects where y developed differently. (I'll describe this in more detail later by quoting the etymologies of specific words.) According to the link that Ricky found (Notes on Etymology by Walter W. Skeat 1901), the digraph "ui/uy" was used in Southern Middle English to represent the sound that developed from older /yː/. Skeat says this is the reason for the spelling of build (which apparently had a long vowel at this point due to homorganic lengthening before the consonant cluster -ld). To support this, he lists the words bruise, from Old English brȳsan, and buy, from a form of Old English bycgan.
I was not able to find any other examples of "ui/uy" being used this way in modern spelling. However, the OED does list muys, muyse as Middle-English spellings of the word mice (from Old English mȳs). For comparison, ice (from Old English īs, without historical rounding) does not have any spellings listed with "ui" or "uy."
Buy seems to be particularly relevant because, like build, it is pronounced with an unrounded vowel in modern English. The OED gives the following information about the etymology:
The forms with the long consonant “cg” seem to have been eliminated through leveling. Another verb that seems to have developed in a similar way is licgan "to lie (down)”: both of these words have the diphthong /aɪ/ in modern pronunciation, which generally develops from Middle English /iː/. But the different spellings suggest that buy may have been pronounced with another vowel at some point, or in some dialects. It's not clear to me from this entry if southwest buy- is the ancestor of the modern pronunciation as well as the spelling (I'm not sure how that development would work in terms of sound changes) or if the modern pronunciation simply comes from midland and northern bie, by.
Google-research suggests that aside from build, buy, and related words, there are no other words where "bui/buy" is pronounced like "bi." (This spelling pattern is discussed in A Survey of English Spelling, by Edward Carney, and Dictionary of the British English Spelling System, by Greg Brooks; both of these sources use the synchronic analysis that "bu" acts as a symbol for /b/.) There are some other words that are superficially similar (buoy, guild, guy, biscuit, conduit, Kuiper) but in general they have different explanations for their spelling, so they are not very useful for explaining build.
However, it might be useful to compare it to busy, for which the OED says:
Another relevant word is bury, from Old English byrgan. According to the Online Etymology Dictionary:
It looks like the modern pronunciation of bury comes from dialects like Kentish, while the spelling comes from dialects like those in the West Midlands.
Build, buy, busy, and bury all have a "b" before the vowel: this is the "labial consonant" mentioned by the OED. The idea that it might have influenced the development of the following vowel in words like these is plausible, since there is strong evidence that it caused the following vowel to retain rounding in words like bull. However, it seems that in some dialects, there were words that retained rounding from Old English y even when it was not directly preceded by a labial consonant, such as bruise, blush, and church.