"He was talking shit, simple as that," Beveridge said.
"In all my time as a coach that's never happened to me. That was totally disrespectful. The guy's gotta learn some damn manners."
Beveridge spoke briefly to Dillard when the pair shook hands at full-time and also let Breakers coach Paul Henare know what he thought of the 27-year-old American's actions.
"Whether it's me or other coaches around here, for him to run down the sideline and get in my face was pretty disrespectful," he said.
"I spoke to him about it after the game so hopefully he realises you've gotta have a little bit of manners in this league. I don't know if his parents would be overly happy with the way he treated an adult."
Beveridge believes the second-placed Hawks (12-11) need to toughen up.
"This is the pointy end of the season and we've gotta have a lot more of an eff you attitude," he said.
"It's nothing but mindset and how you apply yourself to the game."
New Zealand are seeking their fourth straight win when they host Perth on Sunday. "Every game's a big game," Henare said.
"We seem to be finding some form at the right time and we want to try and keep this momentum going."
