If I have hypothesized that a variable $x$ will have a positive effect on $y$, but the analysis shows that the effect is negative, but not significant, how should I interpret these results? Should I say that the results do not support my hypothesis, yet that it is not conclusive whether this is true only for the sample or also for the entire population?
Generally, if a coefficient is statistically insignificant, then we say the estimate is too imprecise to make any claim about suggestive correlation.
However you should do a hypothesis test on the effect of x being positive, not just zero (which is likely what your regression output gave you). This may change your inference.
So $H_0=\beta>0$, with $H_a\leq 0$