I feel like it is important to clarify something here, given the amount of attention this thread is attracting due to the use of the term "Negro", and I'm going to have to do something unusual in commenting both as a flaired user and as a moderator here.

"Negro" is a naturally contentious word in the English language. It is a word richer with historical significance than simply being an out-dated synonym for "black" or "African American", given that there are a number of other words that have been used historically in the United States to describe people of dark skin and/or African descent. In the modern English language, it is of course almost universally and quite rightly demonised as a racially prejudiced phrase. This means that historians are very careful in how they engage with the word "Negro", and with similar kinds of language.

Within history as a discipline, it is absolutely unacceptable to use the term unless it is being used in a very specific historical context - it is fine to use it in quotation of contemporaries, or to reference the idea of 'the Negro' as a trope in contemporary culture and discourse, but absolutely not acceptable to use it as an actual descriptor for people of African descent. In other words: if the words are your own, and not referencing the idea of 'the Negro' in American (or Caribbean etc.) culture, then it is absolutely inappropriate to use the word in an English language publication (even The Journal of Negro History, which clung to the title for nearly a century, is now The Journal of African American History).

This thread did pose quite the challenge as a moderator because its use of the word "Negro" is a little confused. Strictly speaking, the word ought to be used in quotation marks when used in isolation. The term "Negro Spiritual" is, though, a legitimate term, and the slightly misguided use of "Negro" in the second paragraph was probably not intended to cause offence or upset, but to act as a shorthand for "Negro Spiritual". So no, you absolutely should not use the word in isolation and it would not be acceptable within academic history to do so; when you are using the words of someone from the past though (including an out-dated academic's writing), or in reference to a particular historical concept (like "Negro Spiritual"), it would invite harmful anachronism to change the language. Most historians are going to alternate between using 'African American', 'black' and/or 'people of colour' when writing about historic African American experiences in their own words (Afro-American is generally fine, but contentious in some circles, and in general considered to be out of vogue). Similar practice should be followed on AskHistorians.

EDIT: For clarity, because there has been some further confusion - I would generally use the term "African Caribbean" when talking about the British Caribbean, or "African Jamaican" for Jamaica, etc. "African American" is usually best reserved only for the United States, not the Americas at large, and certainly not for people of African descent anywhere in the world.

Surprisingly, there are virtually no primary source documents on Chinese workers on the Transcontinental railroad. If I remember correctly, there are actually researchers simply looking for any primary source material on the topic. The only related primary source documentation I ever found during my brief research on the subject was in Chinese American Voices: From the Gold Rush to the Present by Judy Yung, specifically "Reminiscences of an Old Chinese Railroad Worker." Unfortunately, there's no mention of work songs that I noticed and the account is actually from Canadian railroad workers instead of TCC workers out west. However, it still paints a good picture of the hardships of Chinese railroad workers.

Hao-Hon (who shares his experience) even tells a graphic story of foremen neglect as Chinese workers are sent into a cave with live dynamite:

“Just at that moment the remaining two charges suddenly exploded. Chinese bodies flew from the cave as if shot from a canon”.

Even after witnessing all these horrible things, Hau-hon still asserts that he is “...proud of the fact that we Chinese contributed much to the development of transportation in Canada”. Quite the testament to the Chinese work ethic. I definitely recommend this source for further reading if you're interested on Chinese workers from that era.

In a similar vein, how about Irish work songs? The Irish were often used as indentured servants, so did they have 'work hymn'-type songs?

Unfortunately I am only able to answer part of your question.

I spent some time studying American Folk Music in my undergrad. If you are interested in American folk songs and railroad worker's songs, I recommend "American Ballads and Folk Songs," written by John Lomax and Alan Lomax. This book contains quite a few "negro spirituals" as well as "white spirituals." There are also songs sung by railroad workers, prisoners, cowboys, sailors, and soldiers. Unfortunately, the book was published in the 1930s and it does not contain any specifically Chinese railroad songs. Still, the book may provide a good starting point for you, especially if you can read music. You can find many of its songs in the Library of Congress's collection of folk music recordings. I believe the contents of this book alone prove that the phenomenon of singing work songs is not unique to African Americans within American culture.

I would be interested to see if someone else can provide you with a more relevant source to your question of specifically Chinese railroad songs, but I have personally failed to find accurate collections of American folk songs from the Antebellum period (although later collections often feature songs with roots many hundreds of years old, I have only ever encountered songs in English), and I fear the language barrier may have prevented such collections from ever beginning.

Chinese-American history is not my field of expertise, but I hope I can provide at least some information that might be at least be interesting:

The Chinese Railroad Workers in North America Project at Stanford University has a really wonderful FAQ page that describes some of the difficulties of finding primary sources of time. The project ascribes the dearth of primary sources to a number of reasons: (1) the lack of a comprehensive search, (2) the chaos of Guangdong during the time, and (3) that families might not have considered letters from "lowly" workers to be worth keeping. The FAQ notes, however, that potentially many of the Chinese workers were literate in the modern sense of being able to read and write - they provide a quote by Mark Twain: "the Chinese 'can read, write and cipher with easy facility.'"

That aside, I did some research to see if there were any secondary sources on Chinese railworker songs. Unfortunately, I couldn't find any. The project mentioned before is working on collating testimonials, but the best source is from Judy Yung's book Chinese American Voices: From the Gold Rush to the Present as mentioned by /u/thewonderfulwiz. Not only is there an account from a Chinese worker in Canada, but a short collection of songs from the lonely wives of Chinese male laborers in America. At the least, the presence of these songs hints to the possibility that the male Chinese workers in America had their own work songs.

I did find one tertiary source: From Sea to Shining Sea: A Treasury of American Folklore and Folk Songs by Amy L. Cohn. It's a children's book, but Cohn provides decent notes and references in a lengthy "Notes about the Book" section. Contained is the song "A Gust of Fall Wind," which Cohn claims is a "lament, sung by the Chinese in America." In the "Notes about the Book Section," Cohn states:

Although there is no factual evidence that the Chinese who participated in building the transcontinental railway network sand "A Gust of Fall Wind," the sadness expressed about the coming winter seems to reflect the sorrow of those toiling long hours far from their homes in China.

Cohn does not provide a reference to the claim, so I did some digging to find where this particular poem came from. The essay "An Historical Perspective on the Chinese Americans in American Music Education" by Chee-Hoo Lum and published in UPDATE: Applications of Research in Music Education provides the 1946 publication The Chinese-American Song and Game Book by A. Gertrude Jacobs as the sources of the folk song. The poem can be found on page 56, and is a part of a children's game called "The Grasshopper." Even if the poem was sung by adult Chinese laborers, the song is from northern China - modern day Beijing, and as many of the Chinese railworkers were from southern China.

Ultimately, "A Gust of Fall Wind" is just an example of beautiful Chinese folk poetry - or at least an interesting attempt to interface Chinese and American "cultures," as noted by Jacob in her introduction. However, it cannot be considered an example of a Chinese railworker's song by any stretch of the imagination.

Lum's article "An Historical Perspective" also provides some useful context as to why we might not know of any Chinese "spirituals." Firstly, it was only until 1967 with the "Number 2" principle of the Tanglewood Declaration that educators accepted that "Music of all periods, styles, forms, and cultures belong in the curriculum." According to Lum's research "For a span of 36 years, between 1935 and 1971, no articles related to the music of China or Chinese American heritage appeared in the MEJ [Music Educators Journal]." In general, it seems that the examination of Chinese music, let alone Chinese-American music is still a rather nascent field. I can't seem to find any freely available versions of the article, so I won't upload it, but Lum's examination of the "The Basal Music Series" is also really interesting:

Up to the mid-1960s, the Chinese songs surveyed within the basal series revealed minimal explanations of historical and cultural contexts. As can be seen from Table 2, typically, they would only include a brief general descriptor such as "Chinese work song" or "Chinese river song." In a slightly more detailed descriptor, specific geographical information would be included. Examples include "Song from Hsiu Kiang, China" and "Chinese, Shensi folk tune." (Don't tell anyone I uploaded this pls, it's just a page)

Finally, perhaps the best source of info on this topic you can find is Elissa Sartwell's 2006 dissertation "The Other Side of the Tracks: Railroads, Race, and the Performance of Unity in Nineteenth-Century American Entertainment." In it she focuses in on Chinese-American representation in the realm of performance art, but also delves into "self-representation" for Chinese-Americans:

Records indicate that at any given time, there were at least two Chinese theatre companies presenting traditional Chinese Operas. Significantly, prior to the completion of the transcontinental railroad in 1869, Chinese theatre companies drew an integrated audience – Chinese and white Americans (124)

However, the Chinese music was not well received by white Americans, with an article from 1869 describing the Chinese Opera as being "like ludicrous child's play."

Chinese and Caucasian audiences rarely, if ever, witnessed performances that addressed what it meant to be a Chinese individual living in the United States. Such performances were limited to songs shared with fellow immigrants. A number of these songs were discovered etched in the walls of holding cells on Angel Island, the gateway to San Francisco for ships from China. (126)

Pages 126 to 127 are really worth reading and there is a fascinating analysis of a subset of these prison poems that "tell the stories of the wives left behind in China." From Victor Greene's A Singing Ambivalence:

[T]he hopes and fears of women left behind are expressed by the male arrivals in America. Emigrant men would perform in gender reverse as wives or other females lamenting the loss of family members in works sung abroad . . . [B]y openly expressing themselves as women in song, men used the latter "as a vehicle of their own feelings."

Unfortunately, the Angel Island poetry is date between 1910 and 1940, and were written after the Chinese Exclusion Act in 1886. The impact of this law on Chinese-American perspectives has to be factored in. Also, as Sartwell notes "Yet, these lyrics of resistance were isolated in private spaces, such as jail cells and secret rituals. As such, the self-representation of Chinese Americans was not observed by the mainstream culture, a culture that continued to define "Chinamen" on its stages and in its popular entertainment" (129).

From the other comments made in this thread, it's probably safe to guess that there were indeed worker songs for Chinese-Americans like that of the African-American spiritual, but none of them are yet to be found. I hope my comment has provided some small insight into potentially why this is the case, and the other means in which we can learn about Chinese-American songs and music.