I am making PCB design for Atmega 128. The chip is square in shape and has 16 pins on each side but when I select Atmega 128 from the Atmel Library in Eagle, it comes out rectangular.
I just took a quick look at the Sparkfun and Atmel libraries. All the Atmega 128 packages look square to me. Can you elaborate on what you are seeing? Perhaps you could give us a screen snapshot of the rectangular-ness you are seeing.
You are confusing the schematic symbol with the PCB component.
The symbol has an awful lot of pins that need connecting in the rest of the diagram and convention has it that the symbols are rectangular so that the schematic can be made so it is readable in the right way. It will not necessarily represent the component shape exactly, for instance all input pins may be on one side and all output pins on the other yet in the component they may be intermixed. This helps make the schematic neater and clearer.
Try having a look at the component section of the pert that makes that device - you may find it is the correct footprint.
Put one in a schematic of its own and transfer to to a PCB to see the effect.
and you probably don’t want to make the board exactly like the one you posted - add some decoupling caps on that board nice and close to the pins, rather than having the user of the board require placing them relatively far away.