A 300mm f/2.8 lens using a 1.4X TC will have exactly the same angle of view and DOF as a native 420mm f/4 lens, because that literally is what it becomes.
There is no magic with a TC. For the sake of simple math, imagine a 100mm f/2 lens. This lens has an aperture diameter of 50mm (100mm / 2 = 50mm aperture). Add a 1.4X TC and the focal length changes to 140mm. But the aperture is still 50mm, the TC cannot change that. So the new lens + TC combination is now a 140mm f/2.8 lens, because 140mm / 50mm = 2.8.
The DOF will thus be the same as every other 140mm f/2.8 lens.
Now, back to what your friend was saying, he is fundamentally wrong, but there are some possible ways to misinterpret what he is saying.
For starters, if you use an f/2.8 and an f/4 lens (whether by TC or not) to achieve the same framing, then the faster f/2.8 will give less DOF and the TC has nothing to do with it.
If, on the other hand you compare an f/2.8 shot at 300mm and cropped to the field of view of a 420mm lens as compared to using a 420/4 (by TC) and not cropping, then the DOF will wind up being the same. This is because the shorter focal length precisely offsets the effect of the faster aperture and the greater magnification to deliver the same DOF.
