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) and two objects not yet published in the literature (SN 2019jc and SN 2021ckj). 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 semianalytical model consisting of luminosity inputs from circumstellar interaction and radioactive decay of 56Ni. We infer low ejecta masses (2
M) and 56Ni masses (0.04 M) 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 estimate of 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 its estimated ejecta mass, 56Ni mass, and explosion site properties, we suggest a low-mass, ultra-stripped star as the progenitor of SN 2019jc. For other SNe Icn, we suggest that a WolfRayet star progenitor may better explain their observed properties. This study demonstrates that multiple progenitor channels may produce SNe Icn and other interaction-powered transients.