Why do non-aerobatic aircraft have a negative G limit?
You can get negative load factors (g forces) in different ways than just flying upside down:


Change in pitch: When you push on the control column, the pitch will start to decrease. Depending on how fast you do this, the load factor can even become negative from this. Some aircraft do this intentionally to reduce the g force to exactly zero:


(image source: Wikimedia)

By pushing the control column further forwards, you would get negative gs.
Wind gusts: The limiting factor for airliners1 (which rarely do the maneuver described above) is mostly due to wind gusts and up-/downdrafts. This can (for a short amount of time) result in large changes of the load factor, even into negative gs for a strong gust in the vertical direction. When experiencing strong turbulence, the airspeed should be reduced to turbulent air penetration speed (VB), which reduces the impact of the gusts.


  Turbulent Air Penetration
  
  Severe turbulence should be avoided if at all possible. However, if severe
  turbulence is encountered, use the Severe Turbulence procedure listed in the
  Supplementary Procedures chapter of the FCOM. Turbulent air penetration speeds
  provide high/low speed margins in severe turbulent air.


(Boeing 737 NG FCTM 1.50 General Information)




1 Limits for large aircraft from EASA CS-25:


  (1) The positive limiting load factor
  must not be less than:
  
  (i) 2.5 g with the EFCS
  functioning in its normal mode and with
  the high-lift devices retracted up to
  VMO/MMO. The positive limiting load
  factor may be gradually reduced down
  to 2.25 g above VMO/MMO.;
  
  [...]
  
  (2) The negative limiting load factor
  must be equal to or more negative than:
  
  (i) -1.0 g with the EFCS
  functioning in its normal mode and with
  the high-lift devices retracted;