The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.
Egg cartons might sound contrived but the reals don't necessarily make sense without reference to rulers, scales, etc. And in fact the defining completeness / Dedekind cut conditions for the reals are necessary for doing calculus but any physical interpretation is both pretty abstract and probably false in reality.
Some eggs are smaller than others; some are more dense, etc. Yes, the "count" is maybe sort of interesting in some very specific contexts, but certainly not in any reductive physical context. It only works in an economic context because we have standards like what constitutes a "chicken egg large white grade AAA".