1) He suggested that we should have a party.
2) He suggested we have a party.
3) He suggested that we should not have a party.
4) He suggested we not have a party.
I know 1) 2) 3) are definitely correct, but... what about 4)? Can we omit ' that' and 'should' in that negative sentence?
A minor addendum to 5jj's answer: although formal-sounding in BrE, it is actually quite normal in (even spoken) AmE.
Possibly your subjunctive allergy playing up?