Try looking at http://history.utah.gov/apps/burials/execute/searchburials
They have information on both David and Eliza, you may want to contact Nephi City to see if they have any further information or burial records.
You will also want to contact the Nephi Public Library. They are in the process of putting their card catalog online. http://www.nephi.utah.gov/library/Services.html
The Nephi Chapter of Daughters of Utah Pioneers may also be helpful. They may have a copy of family histories. www.dupinternational.org
(beware: their site is often unavailable) lists some of the copies that are available through the main chapter which is located in the Pioneer Memorial Museum, 300 N Main, Salt Lake City, Utah. The local chapter may be able to put you in contact with living relatives.http://dream.lib.utah.edu/digital/unews/
is the Utah Digital Newspapers project. They are digitizing past newspapers and putting them online. You may find obituaries and marriage announcements. I don't think they have done the Nephi Times News, yet. A very brief search for Cazier came up with an obituary for Samuel Cazier, who had a son named David.
You can find copies of Nephi papers microfilmed by the LDS church by going to www.familysearch.org
and clicking on the Family History Library tab and then searching their catalog: http://www.familysearch.org/eng/Library/FHLC/frameset_fhlc.a...
You will probably want to do a place search and surname search.
It looks like the LDS church has microfilmed both The Ensign (1888 - 1891) and the some of the Nephi Times News. I know the Nephi City Library has copies of back issues, but I don't know how far back they go.
They also have sexton records on microfilm from 1889-1986.
If you are not familiar with the LDS Family History Centers, there are many located throughout the world. There may be one near you. If you are not close enough to visit Salt Lake City, try looking on their website for a Family History Center near you. You can then ask that they have the film sent to that location.
Both BYU and U of U have large collections of Utah history documents and manuscripts. Other colleges in the area also have historical collections. Many colleges are in the process of digitizing those records for preservation purposes and to make them available online.
History of Juab County by Alice P. McCune has a picture of David Cazier on page 62, it may have more information, but there isn't an index.
A History of Juab County by Pearl D. Wilson lists some Caziers (including Samuel), but not a David.
They Left a Record: A Comprehensive History of Nephi Utah 1851 - 1978 by Worthington, Greenhalgh, & Chapman has a lot of references to Cazier including David -- page 21 has an excerpt from his journal.
Hopefully, that gives you a good start!