Am I designing this circuit correctly using Arduino?
A typical green LEDs forward voltage (Vf) is around 2.1V. If you are using a 5V supply then each LED will draw (5V - 2.1V) / 100&Omega; = 29mA. 
So for your 2 LEDs you are asking the poor little I/O pin to supply 58mA, which it's probably won't be too keen on ;-) (I haven't check the ATmega, but typical ratings are ~20mA per I/O pin, although Jippie says it's 40mA so it's maybe more - he's more likely to be right here as I have never used an Arduino. In any case 58mA is still too much)  

Also, a typical maximum current for a 5mm LED is 20mA (can be more - you will have to check your LEDs datasheet), so even if the pin could supply the current it's probably too high for the LED.  

As Chris says, the best way to drive them is to use an external transistor, something like this (the 1k&Omega; value shown can be altered for desired current as described below, the 10k&Omega; is to limit the current through the base of the NPN transistor**):



Also, you could put the LEDs on separate pins and increase the resistor to reduce the current, say to something like 15mA, which would mean a (5V - 2.1V) / 0.015A = 193&Omega; (you can round this up to 200&Omega;) 

** To calculate the base current, you subtract the base-emitter drop from the supply and divide, much like the above calculations:

(5V - 0.7V) / 10k&Omega; = 0.42mA

The thing to make sure of here is that your transistor has enough gain to allow enough current through the LED - say for a gain of 100, we get roughly 0.42mA * 100 = 42mA, so it should be okay.
You could use a bit lower/higher if desired (or you don't have a 10k&Omega;) say from 1k&Omega; to 15k&Omega;  .