Many people’s ancestors arrived on the Mayflower ship that landed in America in 1620. There was no ship called the “Juneflower,” but there are jokes.
 
“I’m always late. My ancestors arrived on the Juneflower” was posted on the newsgroup comp.sys.ibm.pc.misc on December 5, 1993. “She’s always late. Her ancestors arrived on the June flower” was printed in the Vancouver (BC) Sun on August 26, 1997. “I’m always late. My ancestors arrived on the Juneflower” has been printed on many images.
   
   
Wikipedia: Mayflower
The Mayflower was an English ship that famously transported the first English Puritans, known today as the Pilgrims, from Plymouth, England to the New World in 1620. There were 102 passengers, and the crew is estimated to have been about 30, but the exact number is unknown. This voyage has become a cultural icon in the history of the United States, with its story of death and survival in the harsh New England winter environment. The culmination of the voyage was the signing of the Mayflower Compact, an event which established a rudimentary form of democracy, with each member contributing to the welfare of the community.
             
Google Groups: comp.sys.ibm.pc.misc
CD rom for IBM
Forrest Wright
12/5/93
(...)
. OLX 2.1 TD . I’m always late. My ancestors arrived on the Juneflower.
   
Google Groups: alt.fan.jimmy-buffett
Remakes
John Skilton
4/7/94
(...)
* CmpQwk 1.4r8 #134 * I’m always late. My ancestors arrived on the Juneflower
     
Google Groups: rec.humor
Canonical List of Fulldeckisms, 08/01
Alan Silverstein
8/1/97
(...)
Always late…  Her ancestors arrived on the June Flower.
 
26 August 1997, Vancouver (BC) Sun, “City Limits,” pg. D13:
She’s always late. Her ancestors arrived on the June flower.
   
6 May 1998, The Commercial Appeal (Memphis, TN), “C. A. Eye” edited by Jon W. Sparks, pg. C1:
She’s always late. Her ancestors arrived on the Juneflower.
   
Google Books
A Prairie Home Companion
Pretty Good Joke Book
By Brian Becker
Minneapolis, MN: HighBridge
2003
Pg. 83:
Yo’ mama is so slow, her ancestors arrived on the June Flower.
 
Twitter
Marti Lawrence
@Marti_L
Replying to @conniereece
@conniereece - I’m always late.  My ancestors arrived on the June Flower.
1:10 PM - 28 Jul 2007
 
Twitter
My Creative Team
@MyCreativeTeam
My wife’s always late but it’s genetic. Her ancestors arrived on the Juneflower
2:43 PM - 29 Jul 2008
 
Twitter
MeanBot
@MeanBot
Replying to @TaniaArpa
@_tania is always late; her ancestors arrived on the Juneflower.
5:14 AM - 29 Nov 2008
     
Google Books
The Snark Handbook:
A Reference Guide to Verbal Sparring
By Lawrence Dorfman
New York, NY: Skyhorse Publishing
2009
Pg. IX:
She’s always late. Her ancestors arrived on the Juneflower.
 
Twitter
KITT
@k_i_t_t
She’s always late. Her ancestors arrived on the June Flower.
12:00 PM - 14 Feb 2009
 
Twitter
Dougvernx
@Crazyasacoolfox
She’s always late. Her ancestors arrived on the Juneflower.
11:50 PM - 9 Apr 2009
 
Google Books
It’s Better to Die Laughing Than to Be Dead Serious:
How to Be the Life of the party, the Podium, and Everywhere in Between
By Marvin Maupin
Bloomington, IN: AuthorHouse
2010
Pg. 16:
“She’s always late. Her ancestors arrived on the June flower.”
   
Twitter
Isabel Weidner
@IsabelWeidner
I’m always late. My ancestors arrived on the June Flower.
6:10 AM - 15 May 2018