Basically yes. If any one LED has a different voltage drop than its 3 neighbors, the current thru them will be unequal and may lead to failed parts.
The part you selected is not a Logic Level MOSFET, but it should work okay
42m? @ VGS = 4.5V
Check your planned current thru it to make sure it will not overheat.
Having a pulldwn to Gnd on the gate will keep it off during reset/sketch startup when the Arduino outputs are inputs and undefined (i.e. floating).