Context. Debris discs around main-sequence stars indicate the presence of larger rocky bodies. The components of the nearby, solar-type binary α Centauri have metallicities that are higher than solar, which is thought to promote giant planet formation. Aims. We aim to determine the level of emission from debris around the stars in the α Cen system. This requires knowledge of their photospheres. Having already detected the temperature minimum, T min , of α Cen A at far-infrared wavelengths, we here attempt to do the same for the more active companion α Cen B. Using the α Cen stars as templates, we study the possible effects that T min may have on the detectability of unresolved dust discs around other stars. Methods. We used Herschel-PACS, Herschel-SPIRE, and APEX-LABOCA photometry to determine the stellar spectral energy distributions in the far infrared and submillimetre. In addition, we used APEX-SHeFI observations for spectral line mapping to study the complex background around α Cen seen in the photometric images. Models of stellar atmospheres and of particulate discs, based on particle simulations and in conjunction with radiative transfer calculations, were used to estimate the amount of debris around these stars. Results. For solar-type stars more distant than α Cen, a fractional dust luminosity−7 could account for SEDs that do not exhibit the T min effect. This is comparable to estimates of f d for the Edgeworth-Kuiper belt of the solar system. In contrast to the far infrared, slight excesses at the 2.5σ level are observed at 24 µm for both α Cen A and B, which, if interpreted as due to zodiacal-type dust emission, would correspond to f d ∼ (1−3) × 10 −5 , i.e. some 10 2 times that of the local zodiacal cloud. Assuming simple power-law size distributions of the dust grains, dynamical disc modelling leads to rough mass estimates of the putative Zodi belts around the α Cen stars, viz. < ∼ 4×10 −6 M of 4 to 1000 µm size grains, distributed according to n(a) ∝ a −3.5 . Similarly, for filled-in T min emission, corresponding Edgeworth-Kuiper belts could account for ∼10 −3 M of dust. Conclusions. Our far-infrared observations lead to estimates of upper limits to the amount of circumstellar dust around the stars α Cen A and B. Light scattered and/or thermally emitted by exo-Zodi discs will have profound implications for future spectroscopic missions designed to search for biomarkers in the atmospheres of Earth-like planets. The far-infrared spectral energy distribution of α Cen B is marginally consistent with the presence of a minimum temperature region in the upper atmosphere of the star. We also show that an α Cen A-like temperature minimum may result in an erroneous apprehension about the presence of dust around other, more distant stars.