  We study relationships between various set theoretic compactness principles,
focusing on the interplay between the three families of combinatorial objects
or principles mentioned in the title. Specifically, we show the following. (1)
Strong forcing axioms, in general incompatible with the existence of indexed
squares, can be made compatible with weaker versions of indexed squares. (2)
Indexed squares and indecomposable ultrafilters with suitable parameters can
coexist. As a consequence, the amount of stationary reflection known to be
implied by the existence of a uniform indecomposable ultrafilter is optimal.
(3) The Proper Forcing Axiom implies that any cardinal carrying a uniform
indecomposable ultrafilter is either measurable or a supremum of countably many
measurable cardinals. Leveraging insights from the preceding sections, we
demonstrate that the conclusion cannot be improved.
