Hypothesis testing:
I refer to this answer: What follows if we fail to reject the null hypothesis?.
Hypothesis testing is about ''finding statistical evidence for your alternative hypothesis $H_A$'', i.e. whether the data you observe is (statistical) evidence that $H_A$ is true (see What follows if we fail to reject the null hypothesis? for more detail).
So if you want to ''show'' (with the data that you observe) that $p_{high} > p_{low}$ then your alternative hypothesis should be $H_A: p_{high} > p_{low}$ versus $H_0: p_{high} \le p_{low}$. (note that there is no ''$\hat{}"$ here because this is about the ''true'' values for $p_{high}$ and $p_{low}$.
This is a one-sided hypothesis, so this has nothing to do yet with the critical region being one-sided or two-sided. Note that we talk about a critical region, not about a confidence interval. (see below for explanation on confidence intervals).
Once you formulated your hypothesis, you can, using a test-statistic, define a critical region. If the data that you observe i.e. $\hat{p}_{high} - \hat{p}_{low}$ (here is where the "$\hat{}$" comes in, i.e. your data you observe) falls in the critical region then you reject $H_0$ and conclude that your data is evidence in favor of $H_A$. How will we now chose that critical region ? Well, we will, given a significance level $\alpha$ chose that critical region where we can most easily find evidence for $H_A$. In other words, we will chose the critical region with the highest power given $\alpha$. It happens that, for a (univariate) one-sided hypothesis (as you have) a one-sided critical region (i.e. an interval in the tail of the distribution of the test statistic) has more power.
Note that, if you want to show that the ''true'' $p_{high}$ is different from the ''true'' $p_{low}$ then you $H_A^{(1)}$ should be $H_A^{(1)}: p_{high} \ne p_{low}$ and the null should be $H_0^{(1)}: p_{high} = p_{low}$. In that case you're better of with a two-sided critical region.
But your alternative hypothesis is just what you want to show, so if you want to show that $p_{high} > p_{low}$ then this is your alternative hypothesis and your critical region is chosen to have the best power. If you want to show that $p_{high} = p_{low}$ then this should be your alternative (and then you're better of with a two-sided critical region). So it all depends on what the researcher wants to show: do you want to show that ''high treatment has more effect than low treatment'' or do you want to show that ''high and low treatment have a different effect'' ?
Confidence intervals
Confidence intervals are different from critical regions, see Why is there a need for a 'sampling distribution' to find confidence intervals? for explanation on confidence intervals.
Hypothesis testing use the observed data to find 'statistical evidence' for your hypothesis $H_A$ that is just a statement about the ''true parameters'' $p_{high}$ and $p_{low}$ (no $\hat{}$ because it is about the ''true'' values of these parameters.
A confidence interval is an interval (at a chosen confidence level) around the observed value $\hat{p}_{high} - \hat{p}_{low}$ (observed therefore the $\hat{}$) in which you are ''confident'' at a certain level that the interval will contain the ''true'' value of $p_{high}-p_{low}$.
Note that there is only one ''true'' value for $p_{high}$ and $p_{low}$ but we do not know these ''true'' values. On the other hand, each time we take a sample (i.e. each time we observe data) you will usually find another value for $\hat{p}_{high} - \hat{p}_{low}$. You want to use now one such an observation $\hat{p}_{high} - \hat{p}_{low}$ to make inferences about the (one) ''true'' value $p_{high}$ and $p_{low}$. There are two ways to make inferences: (1) make a statement about $p_{high}$ and $p_{low}$ and then ''check'' whether the observation confirms the statement (at a certain significance level), this is hypothesis testing and (2) use the observed data $\hat{p}_{high} - \hat{p}_{low}$ to construct an interval that contains (at a certain confidence level) the true value $p_{high}-p_{low}$. This is a confidence interval.
Finally note that hypothesis tests can be used to construct confidence intervals and vice versa.
