The problem is that array is an array of strings, not numbers. This means that 0 is being turned into a string "0" and then concatenated with other string variables.
You could turn array into an array of Numbers like this:
array = argv.slice(1).map(Number)
Note that I have removed the first entry, the name of the program, using slice.
You can now loop through array starting from element 0 and your code will work as expected.
Working example (making use of the esoteric --> operator ;)
var array = argv.slice(1).map(Number);
var count = array.length;
var sum = 0;
console.log("There are " + count + " individual numbers.");
while (count --> 0) sum += array[count];
console.log(sum);
update
Instead of using map, you may as well use reduce:
var array = argv.slice(1);
console.log("There are " + array.length + " individual numbers.");
var sum = array.reduce(function(prev, curr) { return prev + +curr }, 0);
console.log(sum);
reduce combines all of the values of array using the function provided. The unary + operator is used to convert the values in array to numbers. The second argument is the initial value of the sum.