PHP eregi is super-greedy and it isn't a "whole string match". That means that .* will eat the rest of the line the moment you put it in. What you need to do is this:
eregi('<p><a href=([^>]*)>', $this->client->results, $regs);
That way, it actually terminates the greedy character grabbing at the > character, and $regs[1] will be equal to the URL.
Even then, there are some minor issues. Quotes will come along, so your string will probably be:
"http://foo.com"
instead of just
http://foo.com
Also, you aren't stripping other tags out, so if you parsed <A HREF="http://www.foo.com" onMouseOver="window.open('http://www.pr0n.com');">,
the string you would get would include the mouseover. Bad.
Sorry if this is oddly formatted, btw -- preview isn't working for me, so I can't tell wtf its going to look like 😛