Many thanks to Debra Wilson, who has solved the mystery of the Loyal Americans by finding the following pin:
which bears a striking resemblance to the one I found:
The top pin was associated with the Loyal Americans of the Republic in Springfield, Illinois; the name of the organization was changed to Loyal Americans in 1915. The following year, the name was changed to Loyal American Life Insurance, and then Loyal American Life Association in 1917. In 1934, the Loyal American Life Association merged with the Ben Hur Life Association.
The Loyal Americans of the Republic was incorporated and commenced business on November 7, 1896 in Springfield, Illinois, with E. J. Dunn as its president, and H. D. Cowan as secretary.
The name on the pin would then date it to about 1915. Some of the other items found in the box belonged to the Lair family (or their descendants) of Princeville, Illinois. Based on these facts, I would speculate that the pin belonged to Lawson F. Lair (1833-1923), of Princeville. I might possibly have belonged to his son, William L. Lair, but because I have never seen any of Will’s belongings in my family’s possession, I would tend to think it belonged to Lawson. However, Lawson’s probate file does not mention him having any insurance at the time of his death in 1923.
Again, thank you, Deb!
Sources (besides Debra): http://bulletin.lifeguide.com/issuer_tracing.html Fourth Annual Insurance Report of the Ninth Biennial Period by the Commissioner of Insurance of the State of North Dakota for the Year Ending December 31, 1906 (p. 464)