No, but the answer could depends on how you set up the lead screws, what kind of lead screws are used and how they connect to the stepper. Also it is important how you transfer the linear Z motion into the X gantry.
Note that regularly used TR08 (Trapezoid, 8 mm) lead screws are stiff enough to not flex under the limited load of an X gantry. So, putting them upright or down should not make a difference. Furthermore, the whole X-gantry is usually constraint in movement by the use of linear guide rods or rails.
Note that lead screws are not perfect, certainly the cheaper versions (ball screws are better in that respect). The imperfections can cause the lead screw nut to move in the X-Y direction. If the nut is firmly attached to the X-gantry, this movement can be seen in the print like e.g. waves in Z-direction we call Z-wobble. There are designs like these or these to mitigate that problem. Leaving one end free (so not constraining both ends) of the lead screw is also a mitigation as with constraining both sides, you over-constrain the screw. Note that the spring couplers you see applied frequently are great for misalignment of stepper motor shaft and lead screw, but are detrimental for the Z-axis. When using such couplers for shaft misalignment, you should constrain at least motion in Z-direction.
Furthermore, the whole X-gantry is usually constraint in movement by the use of linear guide rods or rails.Basically the movement is limited in X-Y, only the lead screw imperfections can push it sideways, not the buckling of the screw. $\endgroup$