AA5 pilot reported traffic making a non-standard pattern entry at a non-towered airport resulted in an NMAC on downwind.

Narrative

I was flying back to my home base after a trip for fuel. I announced multiple times my intentions to fly overhead midfield and make a left teardrop to enter the right downwind for Runway XX at ZZZ. As I was making my teardrop turn I hear Aircraft Y announce that he's entering the pattern at the right crosswind leg. I already had a bad feeling hearing a pilot say he's entering the pattern at the right crosswind instead of an approved pattern entry procedure. I enter the downwind at midfield and make my call. Seconds after my call Aircraft Y announces that he is below me and said some other remark that I didn't pay attention to because I looked off my right side and saw Aircraft Y off my right side barely below me and flying in a direction angled towards me and not parallel the runway. I take immediate evasive action; making a climbing steep left turn. As I climb away I noticed he banked towards me more eventually passing under me. He didn't seem to think he did anything wrong. Shortly after he landed; he parked at the FBO; received fuel and departed shortly after. I checked his flight path on FlightAware after he arrived at his destination and it again he joined the pattern on a crosswind leg which is not recommended and potentially dangerous.