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All-star Joe Johnson scored 14 points for the Hawks,who had their three-game road winning streak end on the first night of their five-game West Coast trip.  Atlanta, fourth in the Eastern Conference standings, had won four of five heading into the break.

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1. "end"  is a verb and "have....end" is a phrase,right?

2. what does "heading into the break" mean?

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1. Yes. "To have something" + bare infinitive (or past participle) is a common construction.

They had their luck vanish.

I had my car stolen.

She had her cell phone explode.

2. "The break" is something like "spring break." At some point close to the middle of the season, league play is suspended for a few days. In baseball, they have the All Star Game. In this case, the team will arrive at the break sitting on a good streak, having won four out of their last five games.
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Avangisitting on a good streak
But the Hawks had lost their first game of five West-Coast games,right? Their three-game road winning streak was  ended, so how could the Hawks be "sitting on a good streak"?  I don't understand.
Sorry, I wasn't paying attention to that. I'm not sure exactly where the "break" fits into this sequence.

A three-game road winning streak could be something bizarre like: won 1 game away; lost 5 games at home; won 2 games away; lost 2 games at home; lost 1 game away.

A possible sequence:

(1) won 3 games playing East Coast teams away.

(2) lost 1game and won 2 games at home

(3) lost first game of West Coast trip

(4) won next four games of West Coast trip

(5) BREAK !