I have found a fair amount of information, but I have not yet managed to find any official declaration by Bethesda as to the inspiration for the Atom Cats.
While the sources referenced below collectively assert that the Atom Cats are inspired generically by 1950's or 1970's stereotypical greaser gangs, often characters and groups have character models (for instance, one article put forth James Dean as a potential inspiration). Ultimately, I'm looking for a more specific answer than "greasers"--if such an answer exists--but thus far I have found neither a Bethesda statement or a second-hand reference to such a statement.
- From Fandom (Zeke's Jacket and Jeans): nothing explicitly relevant. Comparable to the more standard Greaser Jacket and Jeans, which is also declared to be worn by Atom Cats. This in turn only reveals that there is at least some measure of Greaser influence, more of which is discussed here, here, and here.
- According to a note regarding the Atom Cats Quest, there is unused content intended to involve the Sole Survivor in a drag race with the Atom Cats. So racing culture may or may not have played a role in inspiring the gang. The racing detail is clarified to have meant power armor racing (which makes sense) in the Fandom page for the Atom Cats Garage.
- On another Fandom page (The Vault - Fallout Wiki > Atom Cats), they are expressly called a 23rd century "greaser gang, except with power armor instead of hot rods". The aforementioned Atom Cats Garage Fandom page has a reference at the bottom, explicitly calling the Atom Cats "a group of friendly Greasers". This alternative Atom Cats Garage page at The Vault - Fallout Wiki did not yield any additional information.
- The two Fandom pages for the Fallout 4 Vault Dweller's Survival Guide transcripts, 1) Locations, and 2) Groups, Gangs, and Factions, did not yield any information on the Atom Cats specifically.
- An IGN article on the Atom Cats Garage states that the Garage "has been renovated into an [sic] 70's style gang hide out."
- One of the aforementioned Atom Cats Garage Fandom pages references Zeke's phrase "Atom Cats Rule!" and states that this is a reference to the Fallout 3 "Tunnel Snakes". Their chosen style is noticeably similar to that of the Atom Cats so I did some investigating here as well; the Tunnel Snakes Fandom page cites their haircuts as being styles from the 1950's. Further in the article under the "Behind the Scenes" heading, it is asserted that "Butch and his gang are modeled after 1950s-style juvenile delinquents known as "Greasers," seen in movies such as West Side Story, The Wild One, Rebel Without a Cause, as well as later "retro" movies and TV programs that hearken back to the same era." Additionally, the Fandom page for the Tunnel Snakes' leader, Butch DeLoria, calls the Tunnel Snakes a "greaser-type gang".
- Unlike the Tunnel Snakes pages, the Fandom pages for the Atom Cats characters Zeke, Bluejay, Duke, Rowdy, Roxy, Johnny D., Peepers, and Aimin' Andy each did not yield any additional information about the gang's inspiration.
- In an article by The Gamer, Bethesda Announces Official Fallout Greaser Gear, part of the description describes the Atom Cats (as well as the Tunnel Snakes and the Kings) as follows: "Fallout’s ‘50s-inspired aesthetics would just be missing something without the greaser-inspired gangs, who look like they’re straight from a James Dean movie...these gangs embodied the cool aesthetic of ‘50s bad boys, complete with coifed hair and leather jackets with their logos on the back."
- An additional site that I found only echoed other sources: "They seem to be a reference to mid-1950s Greasers."
- The FextraLife page for the Atom Cats--as well as the only character page, for Rowdy--appears to be in progress and did not yield any useful information.
- This Bethesda page for Fallout 4 did not have any Atom Cat information.
- Bearing a similar name and style to the Atom Cats, the Stray Cats are an American rockabilly band formed in 1979 with musical inspiration drawn from 1950s artists such as Eddie Cochran, Carl Perkins, Gene Vincent, and Bill Haley & His Comets. More info on the Stray Cats can be found here and here.
- At about 1:40 in FudgeMuppet's YouTube video Fallout 4 - 5 Atom Cats Facts, the video's narrator asserts much the same (that the inspiration for the Atom Cats are inspired by 1950s greasers) though he does not offer more than observation to support this.
Unfortunately, none of these leads have brought me any closer to an official statement from Bethesda as to the inspiration for the Atom Cats.
A Note on the Question Itself
I checked the ARQADE question guidelines here, here, and here. I think the question is suitable, as it asks about characters in a game and the question has a limited scope for a correct answer (Bethesda statement(s), or reference to such). The part that I am uncertain about is this clause: "...site policies prohibit ...Speculations on...developer intent on mechanics and narrative." To be clear, I don't want speculation. Instead, I want matter-of-fact "Bethesda says regarding the Atom Cats' inspiration, this"--I've already seen plenty of observation and assertion regarding the Atom Cats' inspiration, but nothing officially stated by Bethesda. However, I am still uncertain as to whether this question would be considered a violation of policy. If such is the case, I will voluntarily remove the question at the request of a moderator.
Why do I think there may be more to the Atom Cats' inspiration than generic greasers?
One of the things that I noticed is that every single Atom Cat member is a named character--none of them are generically "named". In contrast, there are generic Gunners, Raiders, Rust Devils, and others, each representing minor factions. Given the unique attention given to the Atom Cats, this suggests to me that they may have initially had a more developed story and, consequently, more specific inspiration.
The following are additional details that allude to specific, real-life persons being used as inspiration for named characters. Unless otherwise stated, these references come from the "Behind the Scenes" section of the respective characters' Fandom page.
Major Named Characters
For Curie, it is stated that "Curie is named after Marie Skłodowska Curie, the physicist and chemist who conducted pioneering research on radioactivity. Curie will acknowledge this when starting the quest Emergent Behavior."
For Dogmeat, one comment states: "Dogmeat is modeled after River, a female German Shepherd, and the pet of Joel and Michelle Burgess." Two additional sub-comments give more detail. The Twitter thread referenced in the second comment can be found here.
For Piper, one comment states: "Piper's interactions with Mayor McDonough seem to be inspired by Bob Woodward's and Carl Bernstein's reporting of the Watergate scandal which eventually led to the resignation of President Nixon. This is further lampshaded by Mayor McDonough's "I Am Not a Synth" speech."
For Preston Garvey, a comment says: "While designing Preston Garvey, character artist Dennis Mejillones used several film characters for reference, including the scientist Miles Dyson (played by Joe Morton) in Terminator 2: Judgment Day." The reference this comment lists is to an interview that can be found on YouTube here. Mention of Preston Garvey's inspiration being drawn from Miles Dyson begins around 45:26.
For Ada, the first comment reads: "Ada is likely named after Ada Lovelace, a real-world mathematician and an early innovator in the field of computer science. She shares her namesake with Lady Lovelace, another Assaultron present at the Easy City Downs." Following up on Lady Lovelace, this one's Behind the Scenes section simply states: "Lady Lovelace's name comes from Ada, countess of Lovelace, who lived in the nineteenth century and is often credited as being the first programmer for her work on the Analytical Engine."
Minor Named Characters
For Jack Cabot, it is explained: "The Cabot family is based on the real world Cabots, one of the "first families of Boston." The first Cabot arrived in Boston from Europe in the year 1700 and since then, many business leaders and politicians have hailed from the family." Continuing: "Jack Cabot's appearance and mannerisms are based on real-life rocket scientist Jack Parsons, known for his experimental work with rocket propulsion in the 1940s and for being a leader of the OTO, a religious occultist group which practices magic and rituals."
For Cito the following claim is made: "Everything about Cito, from having been raised by apes to the way he talks and dresses, is a parallel to Tarzan."
For Big Mack, it reads: "Big Mack is named after "Big Mack" Pryor, who met the staff at Bethesda Studios while battling relapsed leukemia at the National Institute of Health in Bethesda, Maryland. He died on December 27, 2015, aged 21." At first glance I thought it was a name-only inspiration, but I suspect that the reason for the namesake character being a supermutant relates loosely to the FEV as a comparison to chemo therapy, or perhaps even the leukemia itself.
For the otherwise-unnamed Vault-Tec Rep, one comment states: "In The Art of Fallout 4, character gen storyboards in the preproduction chapter describe him as "The rep looks like Fredo Corleone, smooth but obviously nervous.""