Is there a better version for verbosing the output of the euclidean method?

There are a couple of "newbie" things here, yes.

• Your method has an uppercase first character, it should be lower-case.

• Your method is not static, which means that an instance of the class has to be created just to use your method. You can make the method static, and add a private constructor to the class so that it can't be initialized from outside.

• As @skiwi says in a comment, please use English identifier names when you code.

• boolean lepesrolLepesre is used to toggle whether or not the method should log anything to console. This is a bad thing. A method like this should not output anything to the console, it should simply return the result.

• This code:

  if (a < b) {
      r = a;
      a = b;
      b = r;
  }
  

  is used to switch so that a always is the largest variable, and it's using r as a temporary variable. Even though this works, I would use a temp variable here that's only visible in this scope.

  if (a < b) {
      int temp = a;
      a = b;
      b = temp;
  }
  int r = a % b;
  

  Note that here, I put the initial calculation of r after this switch has taken place.

• There is no need to use the else as you return inside the if. You can remove else and reduce the indentation there.

Edit:

In the comments below you asked for a cleaner way for the user to see every step, you can accomplish this by using a callback interface.

Then you can modify your code to:

public static interface EuklidesStepCallback {
    void onStep(int a, int b, int r);
}
public static int gcd(int a, int b, EuklidesStepCallback callback) {
    ...
    while (r != 0) {
        r = a % b;
        if (callback != null) {
            callback.onStep(a, b, r);
        }
        a = b;
        b = r;
    }
    return a;
}   

public static void main(String[] args) {
    EuklidesStepCallback callback = new EuklidesStepCallback() {
        @Override
        public void onStep(int a, int b, int r) {
            System.out.println(a + "=" + a / b + "*" + b + "+" + r);
        }
    };
    System.out.println(gcd(24, 16, callback));
}

If you're able to use Java 8, you can use a lambda for the callback:

public static void main(String[] args) {
    System.out.println(gcd(24, 16, (a, b, r) -> System.out.println(a + "=" + a / b + "*" + b + "+" + r)));
}

This callback system is more flexible in case you'd want to save each of the steps to a List or similar.

You are rightly concerned about logging. System.out.println() is fine during development, but should be taken out before you consider your code to be "complete". Using a boolean flag to trigger verbose execution is a bad idea, since it clutters the function's interface for purposes that have nothing to do with your algorithm. Furthermore, you'll still need a mechanism for switching the printouts on and off, preferably without recompilation.

Fortunately, there is a solution: the Java Logging API.

In your code:

import java.util.logging.*;

public class Euklideszi {

    private static final Logger logger = Logger.getLogger("Euklideszi");

    public static int gcd(int a, int b) {
        …
        logger.fine(a + "=" + a / b + "*" + b + "+" + r);
        …
    }
}

(I've changed the method signature, not only to drop the boolean flag. The function should be static, so that you don't have to create a new Euklideszi() to call it. Also, gcd() is a more useful name. Maybe lko(), if you insist.)

By default, messages at the FINE level are suppressed. To enable them, create a euklideszi.properties file containing:

handlers = java.util.logging.ConsoleHandler
.level = FINE
java.util.logging.ConsoleHandler.level = FINE
java.util.logging.SimpleFormatter.format = %2$s: %5$s%n

Then, run Java using the command-line parameter

java -Djava.util.logging.config.file=euklideszi.properties …

to use that configuration. (There are other ways to configure logging — read the documentation.)

Your algorithm implementation seems to be excessively cumbersome.

For example, the first line is int r = a % b;, but then you discard r by using it as temporary storage space for swapping a and b. Then you compute a % b again.

But the heart of the algorithm is in the while loop. I believe you can discard the if (a % b == 0) special case if you write that loop properly.

public static final Logger logger = Logger.getLogger("Euklideszi");

public static int gcd(int a, int b) {
    // Without loss of generality, let a >= b
    if (a < b) {
        int swap = a;
        a = b;
        b = swap;
    }

    int r;
    while ((r = a % b) != 0) {
        logger.fine(a + "=" + a / b + "*" + b + "+" + r);
        a = b;
        b = r;
    }
    logger.fine("A legnagyobb közös osztó: " + b);
    return b;
}

I am not trying to be rude but just think about your question.

You are asking your question in English, and you want us to review your code. But than your code is not English.

One small thing that I saw: you have have if condition, that print some thing if it's true, in two different places. You can convert it to method.

if (lepesrolLepesre) {
     System.out.println("A legnagyobb közös osztó: " + a);
}

private yourConditionMeaning(boolean condition, int value){
     if (condition) {
          System.out.println("A legnagyobb közös osztó: " + value);
     }
}