Caveat Lector: I have no medical training and this post discusses medical conditions, including those that suppurate, necrotize, and/or ooze. The medical aspects of the following post are, therefore, potentially both disgusting and unreliable.
A boil is a bacterial infection of a hair follicle resulting in a painful swollen area on the skin.  Four of the eight missionaries in this study report a total of ten instances of missionaries with boils.  I have not cataloged other types of skin lesions, but blisters and bug bites also appear in the diaries and affected missionaries in similar ways. 
Elder Clark’s experience with a boil, starting 1901 Jul 07, seems to have been the most severe recorded in the diaries. It illustrates most of the difficulties.
“my toe has got a boil on it so I can’t get my Shoe on the man give me one to put on it is painful to walk or Sleep. … [next day] … “we go on but I can’t hardly walk the foot is Swollen quite bad we go about 2 miles I Stop at a house where Elder martindale goes to the office after mail … then we try to do Some Tracting but I have to Set down quite often we try to get entertainment but was 3 times the last place I told the man I couldn’t go no further So finally he consented we go to bed but the Bed bugs nearly eat us up and my toe hurting So bad I didn’t Sleep any the morning came we had Some Breakfast then we leave and try to get Back to a friend about 3 miles when half way we had to Stop my foot hurt So bad. we Stay in the Shade of Some trees nearly all day then we resume our journey with considerable difficulty.” (the suffering continues for thirteen more days in the footnote: )
The so-called “travelling Elders” walked almost everywhere, sometimes as much as fifty kilometers per day. Boils on the feet or legs made walking difficult or impossible; I find four instances of lower-extremity boils, which together resulted in twenty-one lost proselyting days. 
Boils and bug bites hurt and/or itched, which interfered with sleep: “I did not rest very good as the itch was bothering me so. And whenever anything would touch any of my large sores I could hardly stand it, as I had a good many and they are awful bad.”  Boils sometimes appeared alongside colds and fatigue. 
Besides interfering with sleep, the hurting and/or itching, well, hurt and itched. All the time.
The only treatments I have noticed are bacon fat for a necrotizing boil and sulphur for (presumed) bug bites.  Some people are particularly susceptible to boils, which partially explains Clarks miserable summer of 1901: “I have been Boiling all Summer.”  Although the life of a travelling Elder was more congenial to boil formation, Sister Cluff also lost four days to a boil on her face. 
The “Southwestern States Mission” series (homepage) examines mission life in (mostly) Texas around 1900.
 Boils are also known as furuncles; a cluster is called a carbuncle. Inflammation and accumulating pus cause swelling, which causes pressure, which causes pain. In some cases, boils can abscess and/or lead to more systemic infections (bacteremia, sepsis, etc).
 Clark reports that at least two of his companions (Bond and Martindale) also had boils. Clark, Cluff, Duffin, and Jones report boils using the word “boil.” Clark uses the word “boil” on 1901 Jul 07 Sun and the word “sore” on 1901 Jul 08 Mon. Jones reports “sores” on his hands and under each arm, 1900 Aug 04 Sat — 29 Wed, that seem like good candidates for boils, though the evidence is better for “red bug” sores. (See Medline: “They [boils] are most common on the face, neck, armpit, buttocks, and thighs”). Clark (Bond) 1900 Jul 24 Tue – Jul 28 Sat; Clark 1900 Jul 28 Sat; Clark, 1901 Jul 07 Sun – Jul 22 Mon; Clark, 1901 Aug 13 Tue; Clark, 1901 Sep 16 Mon – 17 Tue; Clark (Martindale), 1901 Sep 23 Mon – 24 Tue; Cluff, 1905 Feb 20 Mon; Duffin, 1899 Oct 20 Fri; Jones, 1901 Dec 08 Sun.
 I will (probably) deal with blisters directly in my (as of yet hypothetical) post on foot care and injuries. By “bug bites” I mean any direct injury caused by the bite or sting of an insect, arachnid, or myriapod. Contact dermatitis, like that caused by poison ivy, would also fit in with this post, but I have not yet identified any instances of it in the diaries.
 Continuing: “But got there to Mr Caddells Place on Tuesday night he invited us in we Stayed all night with him the next morning we ask him if we could Stay with him a few days he said he would do the best he could if his wife didn’t get down So my companion went in the field and helped the man in the work. while I Stayed around the house and doctored my foot it was So bad I couldn’t Sleep much at night nor Put my foot to the floor but today it Seems a little better I have been to mr D. Caddells 4 days now the toe has turned Black Some of the people think it may turn in to Blood Poison but I am not Scared much I have been putting fat Bacon on it. it is very hot for Such a bad Sore.” (Clark, 1901 Jul 07 Sun, 08 Mon). I’ve omitted the intervening entries, which continue in a similar vein, but Clark did not proselyte again until Jul 22 when “we get Ready to go on our work again So after dinner we Pay $4.00 dollars for the weeks Board and lodgings and bid mr. Rising good Bye and go by Cadells Place and chat with them a while then mrs. caddell got her little Boy to hitch up the horse and cart and take us on our road about 2 miles to Save my foot which is Still Sore then we visit a few Families and leave Some Tracts….” (Clark, 1901 Jul 22 Mon). I found one other instance of blood poisoning: “Found Elder Dana well and all the saints except one sister, the young girl who was married when we were in here before. She run [ran] a nail in her head and blood poison has set in and there is little hope of her recover[y] but it might be imposible with man to heal her but God can do so if he sees fit. After dinner, Elders Shipp and Dana went to P. O. We administered to her and it releaved her some.” (Folkman, 1901 Apr 04 Thu).
 “the next day Elder Bond was So lame he couldn’t walk with a large boil on his ankle So the man hitched up his team and hauled us 7 miles to Mr. Holders place and now we are Staying here doctoring his leg.” (Clark, 1900 Jul 25 Wed); “Elder Bond’s leg is getting a little Better in the afternoon we leave Mr. Holders place and go about one mile over to Sister Estes place and Stay with her. Bond was hauled over in a wagon By Mr. Holders Friday Brother asy [Asay? Estes?] … I have had 2 boils and have got another coming we remain here all night.” (Clark, 1900 Jul 28 Sat); “it Soon began to get warm So we Stayed in the timber to rest as I am not feeling well today I also have a large Boil on my leg it don’t feel good.” (Clark, 1901 Aug 13 Tue); “But Elder martindale had a Boil on his leg So we didn’t go far nor do much visiting I have got one on my neck one has just disappeared I have been Boiling all Summer” (Clark, 1901 Sep 23 Mon); “So on Tuesday the 24 we go on Slowly But could not go far on account of my companion’s Boil We Set in the Timber most all day. So when night came we trug [drug? trudged?] along Seeking lodgings come to 5 houses before we found a place to Stay” (Clark, 1901 Sep 24 Tue).
 Jones, 1900 Aug 17 Fri. As noted above, there is a good chance Jones is referring to “red bug” lesions and not boils. “I did not sleep very good as the sand flies and red bugs were biting me all night.” (Jones, 1900 Aug 09 Thu).
 “My boil and cold together were causing me to feel very miserable.” (Jones, 1901 Dec 08 Sun); “I have a Boil on my neck makes me feel Sick. I have a heavy cold too.” (Clark, 1901 Sep 17 Tue).
 “I have been putting fat Bacon on it” (Clark, 1901 Jul 08 Mon); “After I put the sulphur on me the itch did not bother me so, but the large sores caused me considerable of pain.” (Jones, 1900 Aug 22 Wed).
 Clark, 1901 Sep 23 Mon.
 “Have been confined to my room with a boil on my face since Thursday, now four days. Yesterday I could not go to S.S. or even regular Services in the [illegible: afternoon?]. Letters from home came to me, to cheer me, papers also. And the day did not seem so long afterall. But it is dreary work, staying in bed and so much of the Lord’s work to be done. … My face is so much better today that I shall be able to write tomorrow.” (Cluff, 1905 Feb 20 Mon). The only boil to appear in Duffin’s diary appeared while he was still a travelling Elder: “Have a boil on the back of my neck which is giving me some trouble.” (Duffin, 1899 Oct 20 Fri).