I am not 100% what your friend meant by that , so lets break it down:
If he meant that a R9 390X does not have the necessary performance to manage 144fps on AAA games, then yes, he is correct. No GPU in the market right now can manage 144fps in 2560x1440p , unless you set details to very low. But the 960 is a lot worse that the R9 390X. Generally , you should be able to get about 80-100 fps in normal games and 45-60 fps in AAA , in ultra settings that is.
If he meant that the screen is somehow not compatible with the R9 390X , then he is incorrect. Screens are not hardware bound, except for freesync and Gsync monitors, but your monitor is neither. A 390X will be quite able to work with your monitor.
The phrase "cant display 144hz " is kind of confusing . To begin with, your monitor has a refresh rate of 144hz. That means that it "refreshes" the image you see 144 times per second. Thus , you can only gain in smoothness up until 144 FPS, because after that, your screen has become too 'slow'. The GPU processes data and sends it to your monitor. The stronger the GPU , the more the FPS. But that has nothing to do with your monitors refresh rate. Your monitor will keep doing its thing in 144hz, regardless of hoe many fps your gpu is processing.
Lastly, some advice: I have a R9 390, and I strongly recommend against getting a R9 390X right now. Make no mistake , my GPU is wonderful. Amazing really, for its price. Its just that in a month or two, nvidia and amd will release their new GPUs , which will offer a LOT more performance that current GPUs.Thus, buying a GPU now is not wise , since you will be wasting your money