Though the specific model of thermal cycler is likely not important for accurate amplification of targets from single cells, the manner in which the thermal cycler is set up can adversely impact your PCR reactions.
Specifically, when placing your tubes in the thermal cycler block, place them in a central location and add a skirt or several empty tubes in the four edges of the block. This helps achieve uniform heating of the sample tubes throughout the temperature cycles.
Emulsion instability may be the result when cycling without a skirt or corner tubes and is likely attributable to hot spots within the thermal block, causing the emulsions to break prematurely.
In addition, the thermal cycler needs to be able to heat the lid to 105 ºC and ramp the temperature at between 1 ºC/second – 4 ºC/second.