If you're having difficulties breathing with your chest cold, then no you should not run. There is no point trying to exercise if you're hacking up a lung. Take it easy for a few days. You will not derail any of your long term efforts at good health or weight loss if a miss a couple of days exercise.
Part of living a healthy lifestyle is knowing when it's time to take things back a notch or two.
So, get plenty of rest and push those fluids. When you start feeling better, you can SLOWLY ease back into a routine. don't try to do too much too soon or you risk being sick for longer.
Listen to your body.