Some missionaries in the Southwestern States Mission noted the passing of birthdays and the anniversary of their leaving home.  If there is any pattern in this small sample, it is that missionaries in their forties were somewhat more likely than their younger colleagues to note birthdays.
The missionaries did not always record birthdays, but I detect no pattern to the omissions other than age.  As seemingly always, travelling Elders had it rough:
“The day was spent at Bro. Thorne’s. Was dull and lonesome. I didn’t celebrate my birthday to any extent. The weather was nice and clear.”
“So I spent my 21st birthday wandering over old Texas hunting for the elders. I was tired and footsore.” 
Elder Clark, a travelling Elder in his forties, merely noted the day without comment.  President Duffin and Sister Cluff’s birthday entries include introspection, stock-taking, resolutions, testimony, and prophecy. In other words, they sound like their New Year’s Day entries. 
The two recorded mission-field birthday celebrations included a “lovely dinner” and a “banquet,” respectively, and one had a “large, beautiful cake.”  Duffin and Cluff both record receiving mailed birthday gifts from home.  They also mention birthday celebrations in Mexico and Utah. 
In 1905 the mission formally commemorated the hundredth anniversary of Joseph Smith’s birth. 
Missionaries sometimes noted the birthdays of distant loved ones, but I find no mention of wedding anniversaries.  There are, however, mentions of mission anniversaries or their fractions. Clark, Cluff, and Jones note one- or two-year anniversary of leaving home.  Clark also noted the twenty-two-month and twenty-three-month marks, while Cluff gave a (characteristic) 160-word statement at her six month mark. 
The “Southwestern States Mission” series (homepage) examines mission life in (mostly) Texas around 1900.
 I count 18 birthdays within the time covered by the diaries, of which 7 are noted. The 40+ missionaries (Clark, Cluff, and Duffin) account for 5 of the noted birthdays. (Duffin also provides 5 un-noted birthdays). Recorded birthdays: Brooks (Dec 23) 1900; Clark (May 12) 1900, 1901; Cluff (Jan 15) 1905; Duffin (May 30) 1904, 1905; and Jones (May 19) 1900. Non-recorded birthdays: Brooks 1899; Carling (Aug 17) 1900; Duffin 1900, 1901, 1902, 1903, 1906; Folkman (Aug 07) 1900, 1901; Forsha (Dec 30) 1899; and Jones 1901. Since birthdays are not necessarily celebrated on the exact day, I checked a few days around the precise date. All of the birthdates come from the diaries except those for Carling and Forsha, which I found on online genealogical databases; I am reasonably confident that I found the right people and reliable information, but I didn’t spend much time on it.
 “dull and lonesome.” (Brooks, 1900 Dec 23 Sun); “21st birthday” (Jones, 1900 May 19 Sat).
 “Saturday the 12 my birthday I am 41 years old today.” (Clark, 1900 May 12 Sat); “Sunday the 12 rainy this is my 42 birth day.” (Clark, 1901 May 12 Sun).
 “My forty fifth milestone reached today. In looking back on the years of my life I am led to wonder if I have any more than that half fulfilled my destiny, and will the future bring me joy or sorrow? One thing I am determined on and that is to try to spend my life or what is left of it in trying to do good to my fellow men. … I have been anxious for a letter…but none came. However, Nettie [Cluff’s daughter] remembers my birthday I am sure.” (Cluff, 1905 Jan 15 Sun).
“This is my birth day. I am now forty-four years of age, in the enjoyment of good health, full of vigor, both in mind and body, and feel very happy in the work of the Lord. Spent this, my birth-day in Traveling from Dublan to El Paso, and in getting ready to leave on the Santa Fe for Kansas City. I feel very grateful for the blessings the Lord has bestowed upon me and mine, and with His help I desire to work all the days of my life for the building up of His kingdom upon this earth; and that my children will have first thought in their minds the “kingdom of God and His righteousness. ” If they will do this they will be blessed in their temporal affairs to the extent that they shall never want for bread, and will always have ample means for any enterprises they wish to establish or carry out for the good of their families and the people. And on this, my forty-fourth birth-day, I bear my testimony to all of my family that every principle revealed through Joseph Smith, the prophet, is of God, heaven born, and is designed to lead the children of our Father into the celestial kingdom of God. And it is my desire to live them, and that my children shall live every principle revealed through that Prophet, as God shall open the way for them.” (Duffin, 1904 May 30 Mon).
“This is my forty-fifth birthday. It finds me full of life and vigor and with increasing desires to be a faithful servant of my Father in heaven and a useful man among my fellows while I live on the earth. The Lord has blessed me exceedingly with faithful wives and children, all of which are true and devoted in the work of the Lord. Altho we have had many hard struggles with adversity, I feel that my Father has been very kind to us, and my soul goes out in gratitude to Him. I have a feeling, too, that He is now going to bless us with much of this worlds goods, with which we shall be enabled to do good in building up His kingdom here upon the earth.” (Duffin, 1905 May 30 Tue).
 Sister Cluff’s shared her 45th birthday party with another missionary: “One of the Elders, D. C. Clayton, had his birthday also and Sister Martin made us a lovely dinner and a large, beautiful cake. I cut one side of it and he the other. There was merry conversation, jests and laughter and all enjoyed the dinner.” (Cluff, 1905 Jan 15 Sun). Cluff also attended the birthday party for a local church member: “This afternoon we attended a banquet in the dining room in celebration of the 40th anniversary of Sister Martin’s birthday, and had a very pleasant time with her and a few of her old friends.” (Cluff, 1904 Dec 13 Tue).
 “Inclosed in a letter from John [her ex-husband, I think] was $10.00 for a birthday present. Dear fellow, I was glad to hear from him.” (Cluff, 1905 Jan 23 Mon). “Received also, a nice silk handkerchief from Mary [his first wife] for a birthday present.” (Duffin, 1904 Jun 1 Wed).
 Duffin was in Utah for Mary’s birthday in 1904: “To-day is my dear wife, Mary’s forty-first birthday and we are honoring her by having a few of our friends in, and having all the family together. Besides our own family there were present Sisters Chamberlain and Fowler and two of their daughters. We spent a very pleasant evening.” (Duffin, 1904 Jun 29 Wed); “Last night we were invited to [illegible: celebrate?] the birth of Bro. and Sr. L. little son and enjoyed it very much.” (Cluff, 1905 Aug 25 Fri); “To-day Amelia [his second wife] and self with a number of others were invited to attend a reception at the home of Apostle John W. Taylor, in honor of his forty seventh birthday. A most enjoyable time was spent.” (Duffin, 1905 May 15 Mon).
 “Last Sunday I appointed committees to prepare memorial services at mission headquarters to commemorate the one hundredth anniversary of the birth of the Prophet Joseph Smith, Dec. 23rd. Today I attended Sunday School and meeting at mission headquarters. Spoke on divine mission of the Prophet Joseph.” (Duffin, 1905 Dec 17 Sun).
 Clark and Forsha were married and Duffin had two wives. Cluff was divorced. “Tuesday the 22 jan My wife’s Birthday foggy we resume our labors….” (Clark, 1901 Jan 22 Tue); “This is my dear father’s anniversary—he is today 75 years old. He is so far away—down in Tobassco, Mexico—and I do not know his address or I would love to write him my congratulations. He is well mentally and physically bright and active for his age.” (Cluff, 1905 Mar 20 Mon); “This p. m. at 63 0 took train…for Utah to present business to the Presidency of the Church…and while I am in to keep, suitably, Mary’s birthday on the 29th.” (Duffin, 1904 Jun 23 Thu).
 None of the missionaries I have noticed fixed on the date of setting apart as the anniversary date; all three used the date of departure from home. “just one year ago today Since I left my home in Utah to come to Texas to Preach the gospel.” (Clark, 1900 Dec 05 Wed); “One year today since I left home.” (Cluff, 1905 May 10 Wed); “Two years from home.” (Jones, 1901 Oct 20 Sun). Cluff (Arizona) and Jones (Nevada) spent around two weeks between home and their official setting apart and departure for the field. Clark lived closer and spent less than a day traveling (left home and arrived in SLC on Dec 05, set apart on Dec 06, leave for mission on Dec 07).
 “22 months today Since I left home” (Clark, 1901 Oct 05 Sat); “just 23 months ago today Since I left Utah to go to Texas on my mission.” (Clark, 1901 Nov 05 Tue) [note the slight imprecision: Clark left home on Dec 05 and left Utah on Dec 07]; “Sunday was six months since my mission to the Central States began, and nearly seven since I left home. How rapidly the time has flown and how pleasantly. It is so pleasant a task to learn the principles of the Gospel of our Lord and Savior Jesus Christ, and try to apply them in our daily life as well as in our missionary work. Elder Stratford, who has been absent nearly three months, tells me he sees a wonderful improvement in my knowledge and explanations. This encourages me for I had begun to think there was no improvement in me, that is, visibly. I feel stronger in the gospel and know that every day my testimony is being strengthened, and I feel that I can live a better life, one nearer my Heavenly Father, that I ever did before. It is my honest and sincere prayer that I may do this, and teach my children the right way to go.” (Cluff, 1904 Nov 28 Sun).