Though this question feels a little more like history than biology, since I was able to easily & quickly find information that answers this question, I will provide a response anyway.
What species of bird inspired Nebraska's nickname Bug-Eating State?
After a bit of research, it appears that no species of bird contributed to this nickname. Instead, Homo sapiens are the culprits! =P
According to the Official Nebraska Government Website:
This bug-eater business is said to have originated during the potato-bug and grasshopper times. An eastern man came out here to visit his relatives. On his return they asked him, of course, how things were in Nebraska? It was our worst time, our one year of a double plague, so he answered, 'Oh, everything is gone up there. The grasshoppers have eaten the grain up, the potato-bugs ate the 'taters all up, and now the inhabitants are eating the bugs to keep alive.' Some newspaper man heard it and published it as a good joke, and it stuck to us for a good while.
At the very first meeting of the [Territorial] Pioneers [Association] in May, 1892, a member protested against the nickname, or totem name as I call it, of Nebraskans, namely, 'bug-eaters.' The 'Sons and Daughters of Nebraska,' composed of those born in the state, was organized at the same time, and they were on the street with ribbons and badges of their order, which had printed on them a big bug and the legend, 'Bug-Eaters.' Of course they did not think how it looked or sounded, but did it merely as a joke. A committee was appointed at the time to suggest a more euphonious and appropriate name, but nothing was really done about the matter until our October meeting this year when the name of 'TREE PLANTERS' was suggested and adopted at once, as emblematic, a pleasant sounding name and most appropriate, for Nebraska has planted more trees since she was named and settled than any other state in the Union.
By all means, then, give us the 'Tree Planters,' which is an honor, a glory, a fact and a pleasant reminiscence of the great change that has come to our prairie country in the short time since its first settlement. Help the Pioneers to fix this name in the public mind, and say good-bye to the Bug-eaters forever. -- Jno. A. MacMurphy, Secy. T.P.A., 1921 Farnam St., Omaha, Neb.
Nebraska now carries the official nickname "The Cornhusker State", and has done so since 1945. More information about the history of Nebraska's nicknames can be found here and here.