Absolute enthalpy values have no meaning; they are always heat content with respect to a reference condition. Only enthalpy differences have meaning, as with duties in exchangers for example. When HYSYS and API TDB enthalpy values differ in sign, it just means that the reference point for ideal gas enthalpy is different. The older (~1980) API TDB used liquid at -200 F as the zero reference point, except for ten light gases, for which it was zero for the vapor at zero Rankine (I don't have the current issue in front of me; it may have changed). This avoids negative enthalpies in typical regions of engineering calculations. This has importance in computerized calculations, when in doing enthalpy balances one would home in on temperature until enthalpy error was a negligible fraction of inlet enthalpy. In calculating that fraction error, you don't want to be dividing by a number close to zero; you want the normalizing value to be comfortably removed from zero. Of course, using zero liquid enthalpy at -200 F means extrapolating latent heat well below the freezing point of most compounds and is purely hypothetical. A roughly equivalent zero reference enthalpy basis would be to use, say, 150 BTU/lb for the ideal gas at -200 F. Exactly what HYSYS has used probably is available in their documentation. It could well be zero for the ideal gas at the ice point. This doesn't mean that HYSYS will have enthalpy balance convergence problems. It just means that more care has been taken to address this; perhaps a different normalizing total enthalpy has been used for deciding when the total enthalpy error has become negligible.
One thing that must be considered when using enthalpy differences as duties is that the differences have meaning only when there is no overall composition change, as there would be across reaction, unless all components involved in the reaction have zero reference enthalpy on the same basis. With the 1980 API TDB system, any reaction involving any of the ten light compounds having different zero basis automatically would give a problem. Regardless of zero enthalpy basis used, net heat of reaction can be calculated by getting enthalpy change to bring all reactants to the standard state for which heats of formation are defined, getting the heat of reaction by changes in heats of formation, adding those two heat terms (observing sign), and using the sum to heat the effluent composition from the standard state to effluent conditions. Either the shortfall is heat added or the excess is heat removed from the system. Either way, one can't always take difference between feed and effluent enthalpies as being net heat of reaction; that assumes too much about zero enthalpy basis for the compounds involved.
HTH, although it may have been more than you wanted.
Dick Russell