Hardy Weinberg Clarification
the population tends towards infinity


Well yes, Hardy–Weinberg principle is based on several assumptions one of them being "population size is very(infinitely) large". Another being "Mating is random".

However, even if the population is small your version of calculation does not make much sense biologically.
If we talk about civets (gonochorists) the probability an offspring has two blue eye color alleles is
(probability mother gives him blue eye allele copy)*(probability father gives him blue eye allele copy). 

The two probabilities are not connected. What allele individual gets from mother has no influence over which allele they get from father. This is assured by "Mating is random" assumption.

To calculate the probability we should calculate:

(allele proportion in all possible mothers)*(allele proportion in all possible fathers)

In your example there is no information about sex of individuals, so the best approximation we have for allele proportion in females/males is the overall allele proportion in the whole population (20%) therefore the best calculation is 20%*20% = 0.04


  but doesn't that assume replacement of alleles!


It seems you think that there is some rule that each parent allele copy is used only once in next generation. This is not true. Under random mating some parents can produce many children and some have none. Some allele copies get replicated several times some don't get replicated at all. In large populations the resulting random changes to allele frequency even out. In small populations allele frequency drifts.