Would an Aero L-159 Alca air frame be able to withstand a supersonic engine?
By 'supersonic jet engine', I presume you mean an engine which would make an L-159 capable of supersonic flight.  Could you mount such an engine on this aircraft?  Possibly, but why would you want to?

An L-159 isn't exactly engineered to be a top of the line high performance fighter; its design and intended use is as an advanced subsonic trainer aircraft based upon the earlier L-39 and L-59 airframes.  Its job is to prepare student pilots for more advanced tactical aircraft and has handling qualities similar to fighters while retaining forgiving flight characteristics for new students so they can more easily build time and experience in high performance jets.  In addition, the aircraft is designed to be affordable and easy to maintain for third world air forces.  Supersonic flight never was considered as a requirement for the role by Aero Vodochody.  Therefore a much more expensive and exotic engine is not needed here.