What are the IUPAC names for singly- and doubly-protonated piperazine?
The relevant rule in the current (corrected) version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) is:

P-73.1.1.2 General rule for systematically naming cationic centers in parent hydrides
A cation derived formally by adding one or more hydrons to any position of a neutral parent hydride (listed in Chapter P-2), or whose degree of hydrogenation has been modified (see P-31) is named by replacing the final letter ‘e’ of the parent hydride name, if any, by the suffix ‘ium’, preceded by multiplying prefixes ‘di’, ‘tri’, etc. to denote the multiplicity of identical cationic centers. (…)

Therefore, the preferred IUPAC names are piperazin-1-ium and piperazine-1,4-diium, respectively.