Thanks to one of the members at the GAC Forum, there is a compiled list of the first June Bugs. So I thought I'd post this for future references.
June Bugs started Friday, June 13th at 8:00 PM (eastern time). It started with the premiere of "Blooper Bunny". This short also aired Saturday and Sunday at the same time. Other than "Blooper Bunny" this is the order the cartoons went in.
Turner didn't have acess to 48 hours worth of Bugs Bunny cartoons, so after the last one on this list, Cartoon Network started from the beginning again. I only watched this the entire first round, so if there are any to add, then I would need to know.
The first time through, they showed "Crazy Cruise", unedited twice in a row. After the first round, they replaced the repeat with "Bugs Bunny Gets the Boid". Also, the first time through, they aired about 7/8 of "The Great Piggy Bank Robbery", probably to fill up time until they were to find "Duck Amuck", which came after this. The only edited cartoons were "Fresh Hare," "Tortoise Wins by a Hare," "Hare Ribbin'" "The Big Snooze", "Frigid Hare", "Ballot Box Bunny", "Bugs Bonnets", and "Southern Fried Rabbit". The cartoons were shown in this order:
"Hare-um, Scare-um"
"A Wild Hare"
"Elmer's Candid Camera" (dubbed)
"The Heckling Hare"
"Tortoise Beats Hare"
"Wabbit Twouble" (dubbed)
"Crazy Cruise"
"Bugs Bunny Gets the Boid"
"Case of the Missing Hare" (dubbed)
"Fresh Hare" (dubbed)
"Hold the Lion, Please" (dubbed)
"The Hare-Brained Hypnotist"
"The Wabbit Who Came to Supper"
"The Wacky Wabbit"
"A Corny Concerto"
"Falling Hare" (dubbed)
"Jack Wabbit and the Beanstalk" (dubbed)
"Super Rabbit"
"Wackiki Wabbit"
"Tortoise Wins By a Hare" (dubbed)
"Buckaroo Bugs" (dubbed)
"Bugs Bunny and the Three Bears" (dubbed)
"Hare Force" (dubbed)
"Hare Ribbin'" (dubbed)
"Little Red Riding Rabbit"
"The Old Grey Hare"
"Stage Door Cartoon"
"What's Cookin', Doc?" (dubbed)
"Hare Conditioned" (dubbed)
"Hare Tonic" (dubbed)
"Hare Trigger (dubbed)
"The Unruly Hare"
"Acrobatty Bunny" (dubbed)
"Baseball Bugs"
"The Big Snooze"
"Hair-Raising Hare" (dubbed)
"Hare Remover"
"Racketeer Rabbit" (dubbed)
"Rhapsody Rabbit"
"A Hare Grows in Manhattan" (dubbed)
"Easter Yeggs" (dubbed)
"Rabbit Transit" (dubbed)
"Slick Hare"
"The Goofy Gophers"
"Buccaneer Bunny" (dubbed)
"Bugs Bunny Rides Again" (dubbed)
"Gorilla My Dreams"
"Haredevil Hare" (dubbed)
"Hot Cross Bunny"
"Rabbit Punch" (dubbed)
"Two Gophers from Texas" (dubbed)
"A-Lad-in His Lamp"
"Frigid Hare"
"Knights Must Fall"
"Rebel Rabbit"
"What's Up, Doc?"
"Bunker Hill Bunny"
"The Rabbit of Seville"
"Rabbit Every Monday"
"Ballot Box Bunny"
"Big Top Bunny"
"Rabbit Fire"
"Rabbit's Kin"
"Duck Amuck"
"Bully For Bugs"
"Southern Fried Rabbit"
"Yankee Doodle Bugs"
"Hyde and Hare"
"Beanstalk Bunny"
"Barbary Coast Bunny"
"A Star is Bored"
"Bugs Bonnets"
"Wideo Wabbit"
"Bedevilled Rabbit"
"Bugsy and Mugsy"
"Rabbit Romeo"
"Hare-Way to the Stars"
"Now Hare This"
"Backwoods Bunny"
"Wild and Woolly Hare"
"A Witch's Tangled Hare"
"Rabbit's Feat"
"From Hare to Heir"
"Lighter Than Hare"
"Horse Hare"
"Prince Varmint"
"Bill of Hare"
"The Unmentionables"
"The Iceman Ducketh"
"Dr. Devil and Mr. Hare"
"False Hare"
"The Odor-able Kitty"