Could have an issue if the surge suppressor requires a breaker to protect it and they only want a 15A or 20A breaker. If it can be protected by the HVAC circuit breaker, then it shouldn't be a problem.
Many ground bars allow 2 or 3 wires of the same size and material in the same hole. Usually there is another stipulation that the wires be between $13 and #10 in size. If this just has a screw and not a bar, you may need to install a bar, drill/tap another hole in the enclosure for the 2nd ground, or use the wirenut and pigtail approach.