I do believe you should get a different interval because the first test is:
That is a two-tailed test, so your rejection region cuts off on each end of the distribution. Your second case is:
This is a one-tailed test, so your rejection region now cuts off from one side of the distribution. Regardless, you should use the test statistic to define your decision rule, not the confidence interval. Of course, using the related confidence interval should give you the same information.
As to your question, suppose it was and your confidence interval included a negative value. Then you would be concluding with 95% confidence that you can have a negative z. You have to reject the hypothesis. Get it? The same idea is being applied, but now we're not limiting the scope of our acceptance region to an interval. We're simply looking for whether or not we stay on the given side our hypothesis suggests. In your case, suppose the interval is (1.5, 2). Then this indicates with 95% confidence that z > 1, as desired. You should find a test statistic that fails to reject the null hypothesis.