I agree with Michael Chernick's answer, but think that it can be made a little stronger. Ignore the 0.05 cutoff in most circumstances. It is only relevant to the Neyman-Pearson approach which is largely irrelevant to the inferential use of statistics in many areas of science.
Both tests indicate that your data contains moderate evidence against the null hypothesis. Consider that evidence in light of whatever you know about the system and the consequences that follow from decisions (or indecision) about the state of the real world. Argue a reasoned case and proceed in a manner that acknowledges the possibility of subsequent re-evaluation.
I explain more in this paper:
http://www.ncbi.nlm.nih.gov/pubmed/22394284
[Addendum added Nov 2019: I have a new reference that explains the issues in more detail https://arxiv.org/abs/1910.02042v1 ]
No, you should not use the Mann-Whitney $U$ test in this circumstance.
Here's why: Dunn's test is an appropriate post hoc test* following rejection of a Kruskal-Wallis test. If one proceeds by moving from a rejection of Kruskal-Wallis to performing ordinary pair-wise rank sum (i.e. Wilcoxon or Mann-Whitney) tests, then two problems obtain: (1) the ranks used for the pair-wise rank sum tests are not the ranks used by the Kruskal-Wallis test; and (2) the rank sum tests do not use the pooled variance implied by the Kruskal-Wallis null hypothesis. Dunn's test does not have these problems
Post hoc tests following rejection of a Kruskal-Wallis test which have been adjusted for multiple comparisons may fail to reject all pairwise tests for a given family-wise error rate or given false discovery rate corresponding to a given $\alpha$ for the omnibus test, just as with any other multiple comparison omnibus/post hoc testing scenario.
Unless you have reason to believe that one group's survival time is longer or shorter than another's a priori, you should be using the two-sided tests.
Dunn's test can be performed in Stata using dunntest (type net describe dunntest, from(https://www.alexisdinno.com/stata)), and in R using the dunn.test package.
Also, I wonder if you might take a survival analysis approach to assessing whether and when an animal dies based on different conditions?
* A few less well-known post hoc pair-wise tests to follow a rejected Kruskal-Wallis, include Conover-Iman (like Dunn, but based on the t distribution, rather than the z distribution, implemented for Stata in the conovertest package, and for R in the conover.test package), and the Dwass-Steel-Citchlow-Fligner tests.
Best Answer
With two samples a Kruskal-Wallis is equivalent to a Wilcoxon-Mann-Whitney but without the direction information; so you lose the ability to do a one-sided test.
Some implementations use the exact distribution for small samples with the Wilcoxon-Mann-Whitney but not for the Kruskal-Wallis (yielding not-so-accurate p-values with small samples). R is one example; if both sample sizes are below 50 (and there are no ties) it uses the exact null distribution Wilcoxon-Mann-Whitney, but it uses the chi-squared approximation for the Kruskal-Wallis, sometimes leading to rejection when you should not reject (as in the example below) or failure to reject when you should.
Even at larger samples the results may differ if the Wilcoxon-Mann-Whitney implements a continuity correction (as in R) but the Kruskal-Wallis does not I don't recall offhand seeing any packages that implement a continuity correction with the Kruskal-Wallis (nor is it quite clear how to do this for more than 2 groups nor that it would be a good idea to do so). This is less important, since they're both approximations - neither is the 'correct' answer - but it's still a potential difference in their decisions in cases where the p-value is near the significance level.