I agree with ponsomime,
in this context a native speaker would definitely say "no hard feelings".
"Sick feelings" refers more to feeling unwell, e.g. there are some Google for "a headache and sick feelings".
I would say, however, that it is better English to say "feeling sick".
The question made me curious, I went a-Googleing and found some entries for "sick feelings" being used in a figurative sense. This, however, refers more to worry / fear, which isn't the same as as "no hard feelings", which has the sense of not bearing a grudge. I wouldn't have used it, though.
Hope this helps