Must we break bread and drink wine on any special day?

No. The breaking of bread was instituted on a weekday evening. We are told to keep it “often” 1 Corinthians 11:26; but we are not told how often, or on which day. The early disciples usually kept this ceremony on “the first day of the week” — Sunday Acts 20:7. For most people nowadays, Sunday still seems to be the most convenient day, but in some countries, (like Nepal where Saturday is the weekly holiday) another day will do.