A “gypsy moth Republican” is a moderate Republican from the Midwest or the Northeast (where gypsy moths cause frequent infestations).  The term “gypsy moth” has been cited in print since at least June 1981 to represent the block of moderate-progressive-liberal Republicans, in contrast to the new conservative Republican president, Ronald Reagan.
 
“Gypsy moth” was coined by one-term Congressman Lawrence J. DeNardis (R-Connecticut) as a counter to the “boll weevil” Democrats of the southern states. The term “gypsy moth Republican” has been infrequently used since the early 1980s, replaced by “Rockefeller Republican” and “Republican In Name Only” (RINO).
 
   
Wikipedia: Gypsy moth Republican
A Gypsy moth Republican, informally, is a politically pejorative term used by conservative Republicans to describe a moderate Republican serving in the U.S. House of Representatives who happens to represent a northeastern or midwestern urban area of the U.S. (an area that is also the habitat for the invasive Gypsy moth, an insect that damages trees and agriculture). Congressman Carl Pursell of Michigan was often described as the quintessential Gypsy Moth Republican in the 1980s.

The implication is that the Gypsy moth Republicans do damage to the Republican Party by siding occasionally with the more liberal Democratic Party.
 
Contrast this term with Boll Weevil Democrats (or Blue Dog Democrats), who are characterized as more moderate or conservative Democrats who occasionally side with the Republicans.
 
Wikipedia: Lawrence J. DeNardis
Lawrence Joseph (Larry) DeNardis (b. March 18, 1938) is a former U.S. Congressman and university president.
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Election and congressional service
DeNardis was elected in the 1980 Ronald Reagan landslide as a Republican to the Ninety-seventh United States Congress (January 3, 1981-January 3, 1983), defeating then-State Senator Joe Lieberman to win a seat that had been held for 22 years by Democrat Robert Giaimo.
 
During his term in Congress DeNardis was known as a moderate “Gypsy Moth” Republican. He was narrowly upset in a 1982 mid-term election characterized by voter anger over a poor economy and the strong grassroots campaign by Democrat Bruce Morrison, a Bill Clinton classmate at Yale Law School. Morrison also won a rematch against DeNardis in 1984, with Reagan again at the top of the ticket.
   
21 June 1981, Seattle (WA) Times, “Morrison in spotlight as G.O.P. ‘frosh whip’” by Richard W. Larsen, pg. A17, col. 3:
“And, of course, we now have the so-called ‘Gypsy Moths,’” says Morrison. They include Republicans, including some of “his” freshmen, “who essentially are from the northeastern quarter of the United States and some from the Midwest—regions which are feeling the pangs of, as they say, reverse growth.”
 
27 July 1981, Oregonian (Portland, OR), “‘Gypsy moths’ hold key to Reagan plan” by David S. Broder, pg. A8, col. 3:
They are self-styled “gypsy moths,” an informal caucus of about two dozen Northeastern and Midwestern Republicans, mainly from urban areas, who are quietly parlaying their role as a minority of the minority into a strong bargaining position.
 
The name was coined by one of the members, freshman Rep. Lawrence J. DeNardis, R-Conn. The gypsy moth is as much of a pest to vegetation in New England and the Great Lakes as the boll weevil is in Dixie.
 
But the term also has a political connotation. “The gypsy moth goes through a unique metamorphosis from worm to fly,” DeNardis said, “and I told Michel (House Minority Leader Robert H. Michel of Illinois) that we preferred not to remain as worms but to fly with the leadership—if we could get some help from the administration on the transition.”
 
The leader of the group is Rep. Carl D. Pursell, R-Mich., a third-termer from the Detroit suburbs. About 33 members have attended some meetings, with two-thirds of them actively involved. THere is no ideological test, but most come out of what once would have been called “the Rockefeller wing” of the Republican Party, moderate-progressive-liberal types who really are regarded as “worms” by some of the more doctrinaire of the dominant conservatives.

Google News Archive
16 September 1981, Toledo (OH) Blade, pg. 3, cols. 5-6:
“Gypsy Moths” Girding For Battle With Reagan
Pursell-Led Caucus Fighting Cuts Hurting Northeast-Midwest
By JOHN BLOCK
WASHINGTON—U.S. Rep. Carl Pursell of Plymouth, Mich., said Tuesday that he and the other “Gypsy Moth” Republican congressmen might have to case a major vote against the Reagan administration to show the White House the seriousness of their concern for the economic plight of the Northeast and Midwest.
 
The Gypsy Moths, an informal alliance of about 25 Republican congressmen from the two regions, are getting ready to flex their muscles.
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Inspired By “Boll Weevils”
He admitted that the inspiration for forming the Gypsy Moths was the so-called “Boll Weevils,” a group of southern conservative Democrats who in many instances broke ranks with the party and gave President Reagan his margin of victory on the Administration’s budget and tax cuts.
 
Google News Archive
21 September 1981, Bangor (ME) Daily News, “‘Gypsy Moth Republicans’” by Michael J. McManus, pg. 16, col. 2:
Meeting in a caucus on Tuesday, 20 Yankee Republicans formally elected Rep. Carl Pursell as chairman and Rep. Bill Green of New York as co-chairman of a group jocularly called the “Gypsy Moth Republicans,” after the insect that bothers this region of the country.
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The"Gypsy Moths” will oppose any more cuts in social or economic development programs, and to fight for a vastly larger set of cuts in militaryspending. “Defense can afford the cuts much more than the social program,” said Rep. Green in a letter to the president signed by a number of the “Gypsy Moths.”