First, this code produces that following output for me:

string: <a href="#1">HIT<b>MUST MATCH</b></a> <a href="#2">2</a>
match:  HIT<b>MUST MATCH</b></a>

string: <a href="#1">HIT<a name="MUST NOT MATCH">-</a> <a href="#2">2<
+/a>
NO match!
[download]

This seems like the result you desire.

However, I would advise you do not use regexes to parse HTML, because it is very, very hard to make the regex do it correctly (especially since there can be >s or <s in comments or quoted attribute values.) It would be better HTML::Parser or HTML::TokeParser. Searches of this site should give you many examples of their use.