find() returns if there exists a substring that matches the expression.
For example, you regex will match "3453457a4234456456" because there is
an 'a' in the expression. What you want is match().

Hold on. The minute you get to handling "|" as well as "&" you will
discover that this is not a regular-expression problem, it is a parsing
problem. You will need to implement operator precedence, and REs can't
do that.

I tidied and commented your code. In doing so the primary error jumped
out.

import java.util.regex.*;
public class Logic
  {
  public static void main(String[] args)
     {
     final StringBuffer strb = new StringBuffer();
     for ( int i = 0; i < args.length; i++ )
        {
        strb.append(args[i]);  //Add the command line arguments to
StrinhBuffer
        }
     final String str = strb.toString(); //Change to a string so
Matcher can use it.
     // look for string of the form ---a&a&b&c---
     final String regex = "[a-z]([&][a-z])*";
     System.out.println("command line:" + str);  //Test print to
ensure the string and regex are correct
     System.out.println("regex:" + regex);
     final Pattern p = Pattern.compile(regex,
Pattern.CASE_INSENSITIVE |
                                       Pattern.UNICODE_CASE);
     // scan command string, not the regex.
     final Matcher m = p.matcher(str);
     if ( m.find() )
        {
        System.out.println("Matched");
        // add some more printout to see what was matched.
        final int gc = m.groupCount();
        // group 0 is the whole pattern matched,
        // loops runs from from 0 to gc, not 0 to gc-1 as is
traditional.
        for ( int i=0; i<=gc; i++ )
           {
           System.out.println( i + " : " + m.group( i ) );
           }
        }
     else
        {
        System.out.println("Not Matched");
        }
     }
  }

if by any chance you are trying the find &xxxx; entities, see
http://mindprod.com/products.html#ENTITIES for a canned solution.

in this line Matcher m = p.matcher(regex);, you shoud pass de str, not
the regex!