This has nothing to do with JSON. Your code represents a (javascript) Object literal/initializer.

From that Object you can map the name property of each entry of the nested array from countries.country.

const myObj = { "countries":
 {"country":[
  {"id":"1","name":"Europe","active":"on","dir":"yes"}, 
  {"id":"2","name":"Africa","active":"on","dir":"yes"},
  {"id":"3","name":"North America","active":"on","dir":"yes"}, 
 ]}
};

const countryNames = myObj.countries.country.map( c => c.name );

console.log(countryNames);

Accepted answer saves a step from mine - noted, thanks.

let obj = {"countries":
 {"country":[
  {"id":"1","name":"Europe","active":"on","dir":"yes"}, 
  {"id":"2","name":"Africa","active":"on","dir":"yes"},
  {"id":"3","name":"North America","active":"on","dir":"yes"}, 
 ]}
}
let objArray = obj.countries.country;
let names = objArray.map(item => item['name']);
console.log(names);

Returns an array of names ['Europe', 'Africa', 'North America']

https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/map

let countryNames = data.countries.country.filter(item => item.name !== '')

OR

const countryNames = []
data.countries.country.forEach(item => {
     countryNames.push(item.name)
})

console.log(countryNames)