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We present a sample of Type Icn supernovae (SNe Icn), a newly-discovered class of transients characterized by their interaction with H- and He-poor circumstellar material (CSM). This sample is the largest collection of SNe Icn to date and includes observations of two published objects (SN 2019hgp and SN 2021csp) as well as two objects (SN 2019jc and SN 2021ckj) not yet published in the literature. The SNe Icn display a range of peak luminosities, rise times, and decline rates, as well as diverse late-time spectral features. To investigate their explosion and progenitor properties we fit their bolometric light curves to a semi-analytical model consisting of luminosity inputs from circumstellar interaction and radioactive decay of $^{56}$Ni. We infer low ejecta masses ($\lesssim$ 2 M$_\odot$) and $^{56}$Ni masses ($\lesssim$ 0.04 M$_\odot$) from the light curves, suggesting that normal stripped-envelope supernova (SESN) explosions within a dense CSM cannot be the underlying mechanism powering SNe Icn. Additionally, we find that an upper limit on the star formation rate density at the location of SN 2019jc lies at the lower end of a distribution of SESNe, in conflict with a massive star progenitor of this object. Based on the estimated ejecta masses, $^{56}$Ni masses, and explosion site properties, we favor a low-mass, ultra-stripped star as the progenitor of some SNe Icn. For others, we suggest that a Wolf-Rayet star progenitor may better explain their observed properties. This study demonstrates that multiple progenitor channels may produce SNe Icn and other interaction-powered transients.