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In this work, the martensitic transformation (MT), shape memory effect (SME) and superelasticity (SE) on [001]-oriented single crystals of Ni44Fe19Ga27Co10, Ni39Fe19Ga27Co15 and Ni34Fe19Ga27Co20 (at %) alloys were investigated. The cobalt content determined any observation of thermal-induced thermoelastic B2-L10 MT and the level of minimum critical stresses necessary to form L10-martensite. Ni44Fe19Ga27Co10 single crystals demonstrated thermal-induced B2-L10 MT, low resistance of the B2-matrix to stress-induced B2-L10 MT, which was observed at 25 MPa, a wide SE temperature range from 263 to 473 K and narrow stress hysteresis of 20–50 MPa. An increase in the cobalt content to 15 at % led to the strain glass transition upon cooling and heating instead of B2-L10 MT and an increase in the critical stresses of L10-martensite formation up to 300 MPa. In Ni39Fe19Ga27Co15 single crystals SE was observed up to 350 K with a narrow stress hysteresis of 20–50 MPa. In Ni34Fe19Ga27Co20 single crystals, no thermal-induced or stress-induced MTs were found and in the temperature interval from 200 to 500 K only plastic deformation of the B2 phase occurred.