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Of New-Zealand origin, the derogatory noun milk-bar cowboy designates a young man who, as part of a motorcycle gang, congregates in or around a milk bar.

The noun milk-bar cowboy is composed of:
– the noun milk bar, designating a local general store selling sweets, newspapers, convenience food, etc., as well as milk, milkshakes, and ice creams;
– the noun cowboy, designating a young man who frequents or hangs around in a certain place or establishment for the purpose of meeting his friends (often the other members of a gang), impressing girls, etc.

The social phenomenon that the noun milk-bar cowboy refers to took place in the 1950s—as mentioned, for example, in the following from The Press (Christchurch, Canterbury, New Zealand) of Thursday 14^th May 1981 [page 17, title & column 6]:

‘Milkbar cowboys’ of the ’50s are now into hot-rods
By Garry Arthur

Respectable Christchurch citizens used to tut-tut at what were known in the 1950s as the “milkbar cowboys,” and wonder as they watched them stropping their motorbikes and old cars around Cathedral Square, what on earth they would turn out like when they grew up.

The reason motorcycle gangs congregated in or around milk bars was explained in the following article from the Evening Express (Aberdeen, Aberdeenshire, Scotland) of Friday 11^th February 1955 [page 7, title & columns 4 & 5]:
—Note: Incidentally, this article is the earliest British-English text that I have found in which the noun milk-bar cowboy occurs:

NEW ZEALAND MILK-BAR COWBOYS

The Teddy-boy ¹ problem has spread to New Zealand, with the difference that, down under, the gangs are known as “Milk Bar Cowboys” (writes John Ashwin).
Fashions, it appears, are different, but the outlook is the same in any “Teddy” language—namely to draw attention to yourself.
To this end the “Milk Bar Cowboys,” because they congregate in milk bars which stay open long after the shops are shut, all own motor cycles which they park in scores along the kerbs, disrupting normal traffic.
Often they race their engines in chorus, creating an insufferable disturbance.
Their uniform consists of peaked caps, flying boots and leather jerkins painted with skull and crossbones on the backs.
Residents say that after the meetings break up members go roaring through the streets with open exhausts seeking unlimited acts of hooliganism.

¹ The British-English noun Teddy boy, which dates back to the mid-1950s, designates a young man affecting a style of dress and appearance held to be characteristic of Edward VII’s reign (1901-10), typically a long velvet-collared jacket, drainpipe trousers and sideburns. (Teddy is a pet form of Edward.)

—Cf. also the nouns hooligan, bovver boy and blouson noir.

These are, in chronological order, the earliest occurrences of the noun milk-bar cowboy that I have found:

1-: From a letter to the Editor, by R. Court-Patience, Hon. Secretary of the Johnsonville Motor-Cycle Club, about the planned compulsory wearing of crash-helmets by motorcyclists, published in The Evening Post (Wellington, Wellington Region, New Zealand) of Saturday 24^th February 1951 [page 8, column 8]:

Sir,—Much has been said in the last few months on the subject of protective headwear for motor-cyclists. The New Zealand Road Safety Council has decided to take some positive action to prevent the large numbers of accidents which occur every year.
Motor-cycle clubs have performed years of work in the promotion of road safety […], but unfortunately their existence and activities are practically unknown by the general public who are so easily prejudiced by the actions of the “milk bar cowboy” type of motor-cyclist who causes most of the trouble for everybody else and in the majority of cases has necessitated the possibility of compulsory crash hats, etc.

2-: From Crash Helmets, listing reactions to the planned compulsory wearing of crash-helmets by motorcyclists, published in The Evening Post (Wellington, Wellington Region, New Zealand) of Saturday 3^rd March 1951 [page 8, column 8]:

Complaining that the Road Safety Council is trying to victimise motor-cyclists even further, R. S. Shields, Hon. Secretary of the Ixion Motor-cycle club, contends that the psychological effect of crash helmets may not be all that the council wishes. “The crazy milkbar cowboys,” he writes, “may become all the more reckless when they have protective headgear on, and the safety of the public may be all the more endangered.”

3-: From General News, published in The Press (Christchurch, Canterbury, New Zealand) of Wednesday 5^th September 1951 [page 6, column 4]:

Milk Bar Cowboys

“He is one of several young hooligans who congregate outside a milk bar with their motor-cycles. They are known as milk bar cowboys,” said Senior Patrol Officer P. Lunn, in the Magistrate’s Court yesterday, prosecuting a youth for using a noisy motor-cycle. “These youths zoom along, with open throttles, in front of other vehicles and make a nuisance of themselves. They give decent motor-cyclists a bad name,” he said. Mr Raymond Ferner, S.M., said that, in the circumstances of the offence, the fine would be £2.

The noun milk-bar cowboy was soon borrowed into Australian English. The following, for example, is from The West Australian (Perth, Western Australia, Australia) of Monday 10^th December 1951 [page 9, column 5]:

MENACE OF “MILK-BAR COWBOYS”

Melbourne, Sunday.—Motor cyclists—sometimes known as “milk-bar cowboys”—were responsible for nine out of ten fatal accidents in which they were involved.
The chairman of the Australian Road Safety Council (Mr. T. G. Paterson) said this in a broadcast last night.
[…]
Mr. Paterson said that the term “milk-bar cowboy” was given to irresponsible motor cyclists by a New Zealand judge recently.
They were also called “boof-head bodgies” ² and “temporary Australians.”
They were the small minority of motor cyclists who caused a large percentage of road accidents by their irresponsible riding.

² In Australian English and New-Zealand English, the noun boofhead designates a foolish or stupid person, and the noun bodgie designates a member of a subculture, especially of the 1950s, characterised by unconventional or controversial behaviour and dress, and similar to that of rockers or Teddy boys.